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0. Review of Algebra4h 16m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 19m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Radical Expressions

College Algebra

0. Review of Algebra

Radical Expressions

Previous problemNext problem

3:29 minutes

Problem 59fBlitzer - 8th Edition

Textbook Question

Solve: √x + √(x + 5) = 5

Verified step by step guidance

1

Isolate one of the square root terms. In this case, move

\sqrt{x}

to the right side of the equation:

\sqrt{x + 5} = 5 - \sqrt{x}

.

Square both sides of the equation to eliminate the square root on the left side. This gives:

(\sqrt{x + 5})^{2} = (5 - \sqrt{x})^{2}

.

Simplify both sides of the equation. The left side becomes

x + 5

, and the right side needs to be expanded using the formula

(a - b)^{2} = a^{2} - 2 a b + b^{2}

.

Set the simplified equation equal to zero by moving all terms to one side:

x + 5 - (25 - 10 \sqrt{x} + x) = 0

.

Simplify and solve the resulting quadratic equation in terms of

\sqrt{x}

, and then square the result again if necessary to solve for

x

.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Square Roots

Square roots are mathematical operations that determine a number which, when multiplied by itself, gives the original number. In the equation √x + √(x + 5) = 5, understanding how to manipulate square roots is essential for isolating variables and simplifying the equation.

Recommended video:

Imaginary Roots with the Square Root Property

Isolating Variables

Isolating variables involves rearranging an equation to get a variable on one side by itself. This technique is crucial in solving equations, as it allows you to simplify the problem and find the value of the unknown variable, in this case, x.

Recommended video:

Equations with Two Variables

Squaring Both Sides

Squaring both sides of an equation is a method used to eliminate square roots. By squaring the entire equation, you can transform it into a polynomial equation, making it easier to solve for the variable. However, this step must be done carefully to avoid introducing extraneous solutions.

Recommended video:

Linear Inequalities with Fractions & Variables on Both Sides

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Watch next

Master Square Roots with a bite sized video explanation from Patrick Ford

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Related Practice

Textbook Question

In Exercises 1–14, multiply using the product rule. b⁴•b⁷

Textbook Question

Write ∛64 using exponents and evaluate.

Textbook Question

In this Exercise Set, assume that all variables represent positive real numbers. In Exercises 1–10, add or subtract as indicated. _ _ 7√3 + 2√3

Textbook Question

Determine whether each statement is true or false. If false, correct the right side of the equation. (y^2)(y^5) = y^7

Textbook Question

Write 27^2/3 in radical form and evaluate.

Textbook Question

In Exercises 1–14, multiply using the product rule. x•x³

Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (c) ( 3x )^1/3

Textbook Question

Evaluate each exponential expression in Exercises 1–22. (−2)^6

Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (d) ( 3x )^-1/3

Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (b) ( -3x )^-1/3

Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (a) ( -3x )^1/3

Textbook Question

In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible. (-27)^⅓

Textbook Question

In Exercises 1–20, evaluate each expression, or state that the expression is not a real number. ___ -√ 36

Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (a) -3x^1/3

Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (d) 3x^1/3

Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (c) 3x^-1/3

Textbook Question

In Exercises 4–6, evaluate each algebraic expression for the given value or values of the variable. 6+2(x-8)³, for x=11

Textbook Question

In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible. -16^¼

Textbook Question

In Exercises 1–20, evaluate each expression, or state that the expression is not a real number. ___ √-36

Textbook Question

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √−25

Textbook Question

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product. √6 (4√6 - 3√2)

Textbook Question

In Exercises 1–20, use the product rule to multiply. _ _ ⁴√9 ⋅ ⁴√3

Textbook Question

In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible. (xy)^⅓

Textbook Question

In Exercises 1–20, evaluate each expression, or state that the expression is not a real number. ____ √1/25

Textbook Question

Perform the operation and/or simplify each of the following. Assume all variables represent positive real numbers. √50

Textbook Question

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √144+25

Textbook Question

In Exercises 1–20, use the product rule to multiply. __ ___ √5x ⋅ √11y

Textbook Question

Determine whether each statement is true or false. If false, correct the right side of the equation. (m^2/3)(m^1/3) = m^2/9

Textbook Question

In Exercises 1–38, solve each radical equation. _____ x = √6x + 7

Textbook Question

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product. ³√3 (³√6 + 7 ³√4)

Textbook Question

In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible. (2xy³)^⅕

Textbook Question

In Exercises 1–20, evaluate each expression, or state that the expression is not a real number. ____ -√9/16

Textbook Question

Perform the operation and/or simplify each of the following. Assume all variables represent positive real numbers. 3√xy - 8√xy

Textbook Question

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √25−√16

Textbook Question

Perform the operation and/or simplify each of the following. Assume all variables represent positive real numbers. (2 + √3) (2 - √3)

Textbook Question

In Exercises 1–20, use the product rule to multiply. ___ __ ⁴√6x² ⋅ ⁴√3x

Textbook Question

Match each expression in Column I with its equivalent expression in Column II. Choices may be used once, more than once, or not at all. a. 5^-3 b. -5^-3 c. (-5)^-3 d. -(-5)^-3 A. 125 B. -125 C. 1/125 D. -1/125

Textbook Question

In Exercises 1–38, solve each radical equation. _____ √2x + 1 = x - 7

Textbook Question

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product. ³√x (³√24x² - ³√x)

Textbook Question

Rewrite each expression without the absolute value bars. |√2-1|

Textbook Question

In this Exercise Set, assume that all variables represent positive real numbers. In Exercises 1–10, add or subtract as indicated. _ _ _ _ 6√7 - ³√x + 2√7 + 5³√x

Textbook Question

In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible. 81^3/2

Textbook Question

In Exercises 1–20, evaluate each expression, or state that the expression is not a real number. ____ √0.81

Textbook Question

Write each root using exponents and evaluate. ∛125

Textbook Question

Evaluate each exponential expression in Exercises 1–22. 4^−3

Textbook Question

In Exercises 1–14, multiply using the product rule. (5x³y⁴)(20x⁷y⁸)

Textbook Question

Write each root using exponents and evaluate. ∛216

Textbook Question

Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √(−17)^2

Textbook Question

In Exercises 1–20, use the product rule to multiply. ___ ___ √x+6 ⋅ √x-6

Textbook Question

Simplify each expression. See Example 1. (3y^4)(-6y^3)

Textbook Question

In Exercises 1–38, solve each radical equation. _____ x = √3x + 7 - 3

Textbook Question

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product. (7 + √2) (8 + √2)

Textbook Question

In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals. _ __ √5 + √20

Textbook Question

Simplify each expression. See Example 1. (n^6)(n^4)(n)

Textbook Question

Evaluate each exponential expression in Exercises 1–22. 2^2⋅2^3

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In Exercises 1–14, multiply using the product rule. (-3x⁴y⁰z)(-7xyz³)

Textbook Question

Simplify each expression. See Example 1. (a^8)(a^5)(a)

Textbook Question

In Exercises 1–20, use the product rule to multiply. ___ _____ ⁶√x-5 ⋅ ⁶√(x-5)⁴

Textbook Question

In Exercises 1–38, solve each radical equation. _____ 3x - √3x + 7 = -5

Textbook Question

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product. (4 + √5) (10 - 3√5)

Textbook Question

In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals. __ __ 5√12 + √75

Textbook Question

Write each root using exponents and evaluate. ∜256

Textbook Question

Write each root using exponents and evaluate. ∛-125

Textbook Question

In Exercises 1–38, solve each radical equation. _____ _____ √6x + 2 = √5x + 3

Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

In Exercises 21–38, rewrite each expression with rational exponents. _ √7

Textbook Question

Write each root using exponents and evaluate. - ∛-343

Textbook Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √6x⋅√3x^2

Textbook Question

Simplify each expression. See Example 1. (35m^4n)(-2/7mn^2)

Textbook Question

In Exercises 21–32, simplify by factoring. __ √27

Textbook Question

Simplify each exponential expression in Exercises 23–64. x^−2y

Textbook Question

In Exercises 15–24, divide using the quotient rule. -56a^12b^10c^8/7ab^2c^4

Textbook Question

In Exercises 21–38, rewrite each expression with rational exponents. _ ∛5

Textbook Question

In Exercises 21–32, simplify by factoring. __ √28

Textbook Question

In Exercises 1–38, solve each radical equation. ____ ____ √x - 4 + √x + 4 = 4

Textbook Question

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product. (√2 + √7)²

Textbook Question

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (6^4)^3

Textbook Question

In Exercises 25–34, use the zero-exponent rule to simplify each expression. 6⁰

Textbook Question

In Exercises 21–38, rewrite each expression with rational exponents. ___ ⁵√11x

Textbook Question

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Textbook Question

In Exercises 1–38, solve each radical equation. ____ 2√x - 3 + 4 = x + 1

Textbook Question

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Textbook Question

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (-2x^5)^5

Textbook Question

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Textbook Question

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Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. -(2x^0y^4)^3

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Textbook Question

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent posi-tive real numbers. p^5/4

Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

Simplify each exponential expression in Exercises 23–64. (x^11)^5

Textbook Question

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. -(x^3y^5/z)^0

Textbook Question

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent posi-tive real numbers. -3 √5p³

Textbook Question

In Exercises 33–46, simplify each expression. __ √5²

Textbook Question

In Exercises 25–34, use the zero-exponent rule to simplify each expression. (13y)⁰

Textbook Question

In Exercises 29–44, simplify using the quotient rule. _______ √x²/144y¹²

Textbook Question

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent posi-tive real numbers. -m √2y⁵

Textbook Question

In Exercises 33–44, add or subtract terms whenever possible. 8√5+11√5

Textbook Question

In Exercises 1–38, solve each radical equation. ____ _____ √x + 2 + √3x + 7 = 1

Textbook Question

In Exercises 33–38, express the function, f, in simplified form. Assume that x can be any real number. _______ f(x) = √81(x-2)²

Textbook Question

Find each root. √12²

Textbook Question

Match each expression in Column I with its equivalent expression in Column II. Choices may be used once, more than once, or not at all. See Example 3. a. 6^0 b. -6^0 c. (-6)^0 d. -(-6)^0 A. 0 B. 1 C. -1 D. 6 E. -6

Textbook Question

Simplify each exponential expression in Exercises 23–64. x^14/x^7

Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

In Exercises 1–38, solve each radical equation. 2(x - 1)¹/³ = (x² + 2x)¹/³

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In Exercises 33–46, simplify each expression. _____ √(x−1)²

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In Exercises 35–52, write each expression with positive exponents only. Then simplify, if possible. (-5)⁻²

Textbook Question

In Exercises 21–38, rewrite each expression with rational exponents. __ 2x ³√y²

Textbook Question

Find each root. ∛x³

Textbook Question

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Textbook Question

Simplify each exponential expression in Exercises 23–64. x^30/x^−10

Textbook Question

In Exercises 33–38, express the function, f, in simplified form. Assume that x can be any real number. ___________ f(x) = √5x² - 10x + 5

Textbook Question

Find each root. ⁷√y⁷

Textbook Question

In Exercises 33–44, add or subtract terms whenever possible. √50x−√8x

Textbook Question

In Exercises 33–46, simplify each expression. ____ √36x⁴

Textbook Question

In Exercises 35–52, write each expression with positive exponents only. Then simplify, if possible. -5⁻²

Textbook Question

In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible. 49^-½

Textbook Question

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. -5^-4

Textbook Question

In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. __ √x⁵

Textbook Question

Find each root. ⁶√x^6

Textbook Question

Simplify each exponential expression in Exercises 23–64. (−4/x)^3

Textbook Question

In Exercises 33–46, simplify each expression. _____ −√100x⁶

Textbook Question

In Exercises 35–52, write each expression with positive exponents only. Then simplify, if possible. x²y⁻³

Textbook Question

In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible. 27^-⅓

Textbook Question

Find each root. √25k⁴m²

Textbook Question

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. (1/3)^-2

Textbook Question

In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. ___ √x⁶y⁷

Textbook Question

Find each root. ∜81p¹²q⁴

Textbook Question

In Exercises 29–44, simplify using the quotient rule. ______ ⁴√13y⁷/x¹²

Textbook Question

In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible. 16^-¾

Textbook Question

In Exercises 33–44, add or subtract terms whenever possible. 3√8−√32+3√72−√75

Textbook Question

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. (4x)^-2

Textbook Question

Find each root. ∜(5 + 2m)⁴

Textbook Question

In Exercises 29–44, simplify using the quotient rule. _______ ⁵√64x¹⁴/y¹⁵

Textbook Question

Simplify each exponential expression in Exercises 23–64. (3x^4)(2x^7)

Textbook Question

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. 4x^-2

Textbook Question

In Exercises 45–66, divide and, if possible, simplify. ___ √200 √10

Textbook Question

In Exercises 39–64, rationalize each denominator. 1 ----- ³√3

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. √7 • √28

Textbook Question

Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. -a^-3

Textbook Question

In Exercises 47 - 49, add or subtract terms whenever possible. 7√5 + 13√5

Textbook Question

In Exercises 45–66, divide and, if possible, simplify. __ ³√54 ³√2

Textbook Question

In Exercises 39–64, rationalize each denominator. 10 ----- ³√5

Textbook Question

In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. _____ ³√x³y¹⁷z²

Textbook Question

In Exercises 47 - 49, add or subtract terms whenever possible. 4√72 - 2√48

Textbook Question

In Exercises 47–54, find each cube root. ___ ³√−27

Textbook Question

In Exercises 35–52, write each expression with positive exponents only. Then simplify, if possible. x⁻²/y⁻⁵

Textbook Question

In Exercises 50 - 53, rationalize the denominator. 30/√5

Textbook Question

In Exercises 39–64, rationalize each denominator. 3 ³√ ---- 4

Textbook Question

In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. ______ ³√32x⁹y¹⁷

Textbook Question

In Exercises 47–54, find each cube root. _____ ³√1/125

Textbook Question

In Exercises 35–52, write each expression with positive exponents only. Then simplify, if possible. a⁻⁴b⁷/c⁻³

Textbook Question

In Exercises 45–54, rationalize the denominator. 7/(√5−2)

Textbook Question

In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible. (2xy)^-7/10

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√x² • ⁵√x³

Textbook Question

In Exercises 39–64, rationalize each denominator. 7 ----- ³√x

Textbook Question

In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. _____ ³√(x+y)⁴

Textbook Question

In Exercises 47–54, find each cube root. ________ ³√−27/1000

Textbook Question

In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible. 5xz^-⅓

Textbook Question

In Exercises 53–58, simplify each expression using the power rule. (x⁶)¹⁰

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∜ m² • ∜ m²

Textbook Question

In Exercises 39–64, rationalize each denominator. 5 ³√ ----- y²

Textbook Question

In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. ___ ⁵√y¹⁸

Textbook Question

In Exercises 45–54, rationalize the denominator. 11/(√7−√3)

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. y^8/y^12

Textbook Question

In Exercises 55–58, find the indicated function values for each function. ___ f(x) = ³√x−1; f(28), f(9), f(0), f(−63)

Textbook Question

In Exercises 53–58, simplify each expression using the power rule. (b⁴)⁻³

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 6^4/6^-2

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. 3^¾ ⋅ 3^¼

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. √14 • √3pqr

Textbook Question

Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ³√8

Textbook Question

In Exercises 45–66, divide and, if possible, simplify. ______ √54a⁷b¹¹ √3a⁻⁴b⁻²

Textbook Question

In Exercises 55–58, find the indicated function values for each function. ____ g(x) = −³√8x−8; g(2), g(1), g(0)

Textbook Question

In Exercises 53–58, simplify each expression using the power rule. (7⁻⁴)⁻⁵

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 4r^-3/6r^-6

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. 16^¾ 16^¼

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∛ 7x • ∛ 2y

Textbook Question

In Exercises 45–66, divide and, if possible, simplify. ____ √50xy 2√2

Textbook Question

Simplify each exponential expression in Exercises 23–64. 10x^4 y^9/30x^12 y^−3

Textbook Question

Simplify the radical expressions in Exercises 58 - 62. ∛81

Textbook Question

In Exercises 39–64, rationalize each denominator. 3 ³√ ------- xy²

Textbook Question

In Exercises 59–72, simplify each expression using the products-to-powers rule. (4x)³

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. √9/25

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 16m^-5n^4/12m^2n^-3

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. x^½ ⋅ x^⅓

Textbook Question

Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ⁴√−16

Textbook Question

In Exercises 45–66, divide and, if possible, simplify. ______ ³√250x⁵y³ ³√2x³

Textbook Question

Simplify each exponential expression in Exercises 23–64. (3x^4/y)^−3

Textbook Question

In Exercises 39–64, rationalize each denominator. 5 ------- ⁴√x

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. x^⅘ x^⅕

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. - ∛5/8

Textbook Question

Simplify the radical expressions in Exercises 58 - 62. 4∛16 + 5∛2

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. -4r^-2(r^4)^2

Textbook Question

In Exercises 45–66, divide and, if possible, simplify. ______ ⁵√96x¹²y¹¹ ⁵√3x²y⁻²

Textbook Question

Simplify the radical expressions in Exercises 58 - 62. ∜(32x^5)/∜(16x) (Assume that x > 0.)

Textbook Question

In Exercises 39–64, rationalize each denominator. 10 ---------- ⁵√16x²

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∜m/n⁴

Textbook Question

Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ⁵√(−3)^5

Textbook Question

Simplify each exponential expression in Exercises 23–64. (3a^−5 b^2/12a^3 b^−4)^0

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (5a^-1)^4(a^2)^-3

Textbook Question

In Exercises 45–66, divide and, if possible, simplify. _______ ³√x²+7x+12 ³√x+3

Textbook Question

In Exercises 39–64, rationalize each denominator. 3xy² ----------- ⁵√8xy³

Textbook Question

In Exercises 61–82, multiply and simplify. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. __ ___ √8x ⋅ √10y

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (p^-2)^0/5p^-4

Textbook Question

In Exercises 59–76, find the indicated root, or state that the expression is not a real number. ___ ⁴√−16

Textbook Question

Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ⁶√1/64

Textbook Question

In Exercises 65–74, simplify each radical expression and then rationalize the denominator. 25 --------- √5x²y

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (3pq)q^2/6p^2q^4

Textbook Question

In Exercises 59–76, find the indicated root, or state that the expression is not a real number. ___ ⁵√−1

Textbook Question

In Exercises 59–72, simplify each expression using the products-to-powers rule. (-3x⁻²)⁻³

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Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∜∛2

Textbook Question

In Exercises 65–74, simplify each radical expression and then rationalize the denominator. 150a³ - √ ---------- b⁵

Textbook Question

In Exercises 61–82, multiply and simplify. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. ___ _____ √5xy ⋅ √10xy²

Textbook Question

Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√∛9

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 4a^5(a^-1)^3/(a^-2)^-2

Textbook Question

In Exercises 59–76, find the indicated root, or state that the expression is not a real number. ___ ⁶√−1

Textbook Question

In Exercises 59–72, simplify each expression using the products-to-powers rule. (5x³y⁻⁴)⁻²

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. (2x^⅕)⁵

Textbook Question

In Exercises 65–74, simplify each radical expression and then rationalize the denominator. 5m⁴n⁶ √ ------------- 15m³n⁴

Textbook Question

In Exercises 61–82, multiply and simplify. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. __ _ 3√15 ⋅ 5√6

Textbook Question

In Exercises 59–76, find the indicated root, or state that the expression is not a real number. ___ −⁴√256

Textbook Question

In Exercises 59–72, simplify each expression using the products-to-powers rule. (-2x⁻⁵y⁴z²)⁻⁴

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. (25x⁴y⁶)^½

Textbook Question

Simplify by reducing the index of the radical : [y^3]^(1/6)

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. √192

Textbook Question

In Exercises 59–76, find the indicated root, or state that the expression is not a real number. __ ⁶√64

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. (x^½y^-⅗)^½

Textbook Question

In Exercises 73–84, simplify each expression using the quotients-to-powers rule. (2/x)⁴

Textbook Question

Simplify the radical expressions in Exercises 67–74, if possible. ⁵√64x^6/⁵√2x

Textbook Question

Evaluate each expression. See Example 7. 169^1/2

Textbook Question

In Exercises 65–74, simplify each radical expression and then rationalize the denominator. 15 ------------ ³√-27x⁴y¹¹

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ∛250

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. 3^½ ⋅ 3^¾ 3^¼

Textbook Question

In Exercises 73–84, simplify each expression using the quotients-to-powers rule. (x³/5)²

Textbook Question

Evaluate each expression. See Example 7. 16^1/4

Textbook Question

In Exercises 75–82, add or subtract terms whenever possible. 4⁵√2+3⁵√2

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. - ∜243

Textbook Question

In Exercises 75–92, rationalize each denominator. Simplify, if possible. 15 ---------- √6 + 1

Textbook Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. (3y^¼)³ y^1/12

Textbook Question

In Exercises 73–84, simplify each expression using the quotients-to-powers rule. (- 3x/y)⁴

Textbook Question

In Exercises 77–90, simplify each expression. Include absolute value bars where necessary. __ ³√x³

Textbook Question

Evaluate each expression. See Example 7. (-64/27)^1/3

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. -9 ⁵√243

Textbook Question

In Exercises 75–92, rationalize each denominator. Simplify, if possible. 17 ---------- √10 - 2

Textbook Question

Evaluate each expression. See Example 7. (-4)^1/2

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. __ ⁸√x²

Textbook Question

In Exercises 77–90, simplify each expression. Include absolute value bars where necessary. __ ⁴√y⁴

Textbook Question

In Exercises 75–82, add or subtract terms whenever possible. ³√54xy^3−y³√128x

Textbook Question

In Exercises 75–92, rationalize each denominator. Simplify, if possible. 12 ------------ √7 + √3

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ∛(16 (-2)⁴ (2)⁸)

Textbook Question

Match each expression in Column I with its equivalent expression in Column II. See Example 8. a. (4/9)^3/2 b. (4/9)^-3/2 c. -(9/4)^3/2 d. -(9/4)^-3/2 A. 27/8 B. -27/8 C. 8/27 D. -8/27

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. ___ ³√8a⁶

Textbook Question

In Exercises 77–90, simplify each expression. Include absolute value bars where necessary. ____ ³√−8x³

Textbook Question

In Exercises 75–92, rationalize each denominator. Simplify, if possible. √b ---------- √a - √b

Textbook Question

In Exercises 75–82, add or subtract terms whenever possible. √3+³√15

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. 8^2/3

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. _____ ⁵√x¹⁰y¹⁵

Textbook Question

In Exercises 77–90, simplify each expression. Include absolute value bars where necessary. ____ ³√(−5)³

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. √24m⁶n⁵

Textbook Question

In Exercises 85–116, simplify each exponential expression. x³/x⁹

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ∜(x⁴ + y⁴)

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ∛(27 + a³)

Textbook Question

In Exercises 85–116, simplify each exponential expression. 20x³/-5x⁴

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ⁹√5³

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ⁶√11³

Textbook Question

In Exercises 83–90, evaluate each expression without using a calculator. 16^(−6/2)

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ⁸√5⁴

Textbook Question

In Exercises 75–92, rationalize each denominator. Simplify, if possible. 2√6 + √5 -------------- 3√6 - √5

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ⁶√x¹⁸y²

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (3^1/2)(3^3/2)

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. ____ ⁹√x⁶y³

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (64^5/3)/(64^4/3)

Textbook Question

In Exercises 93–104, rationalize each numerator. Simplify, if possible. 5 √ --- 3

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ⁹√∜7³

Textbook Question

In Exercises 91–100, simplify using properties of exponents. (x^2/3)^3

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (y^7/3)(y^-4/3)

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. 8√(2x) - √(8x) + √(72x)

Textbook Question

In Exercises 85–116, simplify each exponential expression. (2a⁵)(-3a⁻⁷)

Textbook Question

In Exercises 93–104, rationalize each numerator. Simplify, if possible. ³√2x ³√y

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (k^1/3)/(k^2/3)(k^-1)

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. 3√72m² - 5√32m² - 3√18m²

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. ___ __ ⁴√a²b ⋅ ³√ab

Textbook Question

In Exercises 85–116, simplify each exponential expression. (-¼x⁻⁴y⁵z⁻¹)(-12x⁻³y⁻¹z⁴)

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (z^3/4)/(z^5/4)(z^-2)

Textbook Question

In Exercises 93–104, rationalize each numerator. Simplify, if possible. √x + 4 √x

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (x^1/4y^2/5)^20/x^2

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. 2∛3 + 4∛24 - ∛81

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. _ ⁴√x ⁵√x

Textbook Question

In Exercises 85–116, simplify each exponential expression. 6x²/2x⁻⁸

Textbook Question

In Exercises 93–104, rationalize each numerator. Simplify, if possible. √a - √b √a + √b

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. ∜32 + 3∜2

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. __ ³√y² ⁶√y

Textbook Question

In Exercises 85–116, simplify each exponential expression. x⁻⁷/x³

Textbook Question

Evaluate each expression for p=-4, q=8, and r=-10. q+r / q+p

Textbook Question

In Exercises 101–108, simplify by reducing the index of the radical. ⁴√7^2

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. 2∛16 + ∛54

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. __ ⁴√√x

Textbook Question

In Exercises 85–116, simplify each exponential expression. 30x²y⁵/-6x⁸y⁻³

Textbook Question

Evaluate each expression for p=-4, q=8, and r=-10. 3q/r - 5/p

Textbook Question

In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. |−20| □ |−50|

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. 3x∛xy² - 2∛8x⁴y²

Textbook Question

In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. ____ √√x²y

Textbook Question

In Exercises 85–116, simplify each exponential expression. -24a³b⁻⁵c⁵/-3a⁻⁶b⁻⁴c⁻⁷

Textbook Question

Evaluate each expression for p=-4, q=8, and r=-10. 5r / 2p-3r

Textbook Question

In Exercises 105–110, use an associative property to write an algebraic expression equivalent to each expression and simplify. 4+(6+x)

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (z^1/3z^-2/3z^1/6)/(z^-1/6)^3

Textbook Question

In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. 30/40−3/4 □ 14/15⋅15/14

Textbook Question

In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (x^−2 y)^−3/(x^2 y^−1)^3

Textbook Question

Evaluate each expression for p=-4, q=8, and r=-10. q/2-r/3 / 3p/4+q/8

Textbook Question

In Exercises 101–108, simplify by reducing the index of the radical. ⁹√x^6 y^3

Textbook Question

In Exercises 101–108, simplify by reducing the index of the radical. ¹²√x^4y^8

Textbook Question

Evaluate each expression for p=-4, q=8, and r=-10. -(p+2)²-3r / 2-q

Textbook Question

Calculate each value mentally. (0.25^3)(400^3)

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. ∛64xy² + ∛27x⁴y⁵

Textbook Question

Calculate each value mentally. (24^2)(0.5^2)

Textbook Question

In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (3x−4 y z−7)(3x)−3

Textbook Question

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (49x^−2y^4)^−1/2(xy^1/2)

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. ∜81x⁶y³ - ∜16x¹⁰y³

Textbook Question

Calculate each value mentally. 15^4/5^4

Textbook Question

Calculate each value mentally. (0.2^2/3)(40^2/3)

Textbook Question

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (x^−5/4y^1/3x^−3/4)^−6

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. 5√6 + 2√10

Textbook Question

In Exercises 85–116, simplify each exponential expression. (x⁻⁵y⁸/3)⁻⁴

Textbook Question

In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (2^−1x^−3y^−1)^−2(2x^−6y^4)^−2(9x^3y^−3)^0/(2x^−4y^−6)^2

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. √6(3 + √7)

Textbook Question

In Exercises 85–116, simplify each exponential expression. (20a⁻³b⁴c⁵/-2a⁻⁵b⁻²c)⁻²

Textbook Question

Calculate each value mentally. (20^2/3)/(5^3/2)

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. 4√3(√7 - 2√11)

Textbook Question

In Exercises 117–124, simplify each exponential expression. 9y⁴/x⁻² + (x⁻¹/y²)⁻²

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. (√2 + 3) (√2 - 3)

Textbook Question

Make Sense? In Exercises 119–122, determine whether each statement makes sense or does not make sense, and explain your reasoning. ____ ⁴√(−8)⁴ cannot be positive 8 because the power and the index cancel each other.

Textbook Question

In Exercises 117–124, simplify each exponential expression. (3x⁴/y⁻⁴)⁻¹(2x/y²)³

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. (∛11 - 1) (∛11² + ∛11 +1)

Textbook Question

In Exercises 117–124, simplify each exponential expression. (-4x³y⁻⁵)⁻²(2x⁻⁸y⁻⁵)

Textbook Question

___ The domain of f(x) = ³√x−4 is [4, ∞).

Textbook Question

In Exercises 117–124, simplify each exponential expression. (2x²y⁴)⁻¹(4xy³)⁻³ / (x²y)⁻⁵(x³y²)⁴

Textbook Question

_ If x=−2, then √x⁶ = x³.

Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. (3√2 + √3) (2√3 - √2)

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. 5/√2

Textbook Question

Between which two consecutive integers is -√26? Do not use a calculator.

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∛2/3

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∜2/25

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. √2/3x

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. √(x⁵y³)/z²

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∛8/x⁴

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. (∛mn • ∛m²) / ∛n²

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. (-4 /∛3 ) + (1/∛24) - ( 2/∛81)

Textbook Question

Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ( 5/∛2) - ( 2/∛16) + (1/∛54)

Textbook Question

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 1/(2 + √5)

Textbook Question

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (√7 - 1) / (2√7 + 4√2)

Textbook Question

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (p - 4) / (√p + 2)

Textbook Question

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 3m / 2 + (√m + n)

Textbook Question

Concept Check: By what number should the numerator and denominator of 1/(∛3 - ∛5) be multiplied in order to rationalize the denominator? Write this fraction with a rationalized denominator.

Textbook Question

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (b) -3x^-1/3

Textbook Question

Write each root using exponents and evaluate. ∜81

Textbook Question

If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent posi-tive real numbers. -12x^1/2

Textbook Question

Find each root. √(-12)²

Textbook Question

Solve: 2x^(2/3) - 5x^(1/3)-3 = 0.

Textbook Question

Solve for x: ∛(x√x) = 9

Textbook Question

Solve without squaring both sides: 5 - (2/x) = √(5 - 2/x).

Textbook Question

In Exercises 101–106, solve each equation. |√x - 5| = 2

Textbook Question

Solve each radical equation in Exercises 11–30. Check all proposed solutions. √(1 + 4√x) = 1 + √x

Textbook Question

Solve each equation for the specified variable. (Assume all denominators are nonzero.) d=k√h, for h

Multiple Choice

Evaluate the radical.

$-\sqrt{\frac14}$

Multiple Choice

Evaluate the radical.

$\sqrt{\left(-5\right)^2}$

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