0. Review of Algebra

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

Write ∛64 using exponents and evaluate.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

Write each root using exponents and evaluate. ∛125

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

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

Write each root using exponents and evaluate. ∛216

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Write each root using exponents and evaluate. ∜256

Write each root using exponents and evaluate. ∛-125

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

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product. (7 - 2√7) (5 - 3√7)

In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals. ___ ___ 5√45x - 2√20x

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

Write each root using exponents and evaluate. ∛-343

Write each root using exponents and evaluate. ∜-81

In Exercises 15–24, divide using the quotient rule. 15x⁹/3x⁴

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √10x⋅√ 8x

In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals. __ __ 3³√24 + ³√81

Write each root using exponents and evaluate. ∜-256

Evaluate each exponential expression in Exercises 1–22. 3^8/3^4

Write each root using exponents and evaluate. ⁵√32

In Exercises 15–24, divide using the quotient rule. x⁹y⁷/x⁴y²

Simplify each expression. See Example 1. (5x^2y)(-3x^3y^4)

In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible. (xy)^4/7

In Exercises 1–20, evaluate each expression, or state that the expression is not a real number. ______ √16 − 25

Evaluate each exponential expression in Exercises 1–22. 2^3/2^7

In Exercises 15–24, divide using the quotient rule. 50x²y⁷/5xy⁴

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In Exercises 21–32, simplify by factoring. ___ √40x

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

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

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

In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals. ______ ___ √4x - 12 + √x-3

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. -5z^2/3

In Exercises 21–38, rewrite each expression with rational exponents. __ √x³

In Exercises 21–32, simplify by factoring. __ ³√54

In Exercises 1–38, solve each radical equation. (3x - 6)¹/³ + 5 = 8

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

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

In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals. ___ ___ 4³√x⁴y² + 5x³√xy²

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

Simplify each exponential expression in Exercises 23–64. x^−5⋅x^10

In Exercises 21–32, simplify by factoring. _____ ³√250x³

In Exercises 1–38, solve each radical equation. (2x + 3)¹/⁴ + 7 = 10

In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product. (3 - 5√2) (3 + 5√2)

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (p^4/q)^2

In Exercises 29–44, simplify using the quotient rule. _____ √19/25

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. (5r + 3t)^4/7

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √24x^4/√3x

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. ⁵√ k²

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

Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (-5n^4/r^2)^3

In Exercises 29–44, simplify using the quotient rule. _____ ³√11/64

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √500x^3/√10x^−1

In Exercises 1–38, solve each radical equation. (x - 2)¹/² + 8 = 6

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

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

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

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

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

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

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

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

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

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

Find each root. √12²

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

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

In Exercises 33–46, simplify each expression. ____ √(−4)²

In Exercises 35–52, write each expression with positive exponents only. Then simplify, if possible. 3⁻²

In Exercises 21–38, rewrite each expression with rational exponents. ____ (⁶√7xy² ) ⁵

In Exercises 33–44, add or subtract terms whenever possible. 4√13x−6√13x

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

In Exercises 33–38, express the function, f, in simplified form. Assume that x can be any real number. _______ f(x) = ³√48(x-2)³

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

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

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

Find each root. ∛x³

In Exercises 1–38, solve each radical equation. (x - 2)¹/⁴ = (3x - 8)¹/⁴

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

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

Find each root. ⁷√y⁷

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

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

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

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

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

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

Find each root. ⁶√x^6

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

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

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

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

Find each root. √25k⁴m²

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

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

Find each root. ∜81p¹²q⁴

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

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

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

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

Find each root. ∜(5 + 2m)⁴

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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)

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

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

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

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)