Table of contents
- 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
Problem 15
Textbook Question
Write each root using exponents and evaluate. ∛-125
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1
Identify the expression: ∛(-125) is asking for the cube root of -125.
Recall that the cube root of a number is the same as raising that number to the power of 1/3.
Express the cube root using exponents: (-125)^{1/3}.
Recognize that -125 can be rewritten as (-5)^3, since (-5) * (-5) * (-5) = -125.
Apply the exponent rule: ((-5)^3)^{1/3} = (-5)^{3 * (1/3)} = (-5)^1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Roots and Exponents
Roots are the inverse operations of exponents. The nth root of a number 'a' is a value 'b' such that b^n = a. For example, the cube root of -125 can be expressed as -125^(1/3), which indicates that we are looking for a number that, when raised to the power of 3, equals -125.
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Rational Exponents
Negative Numbers and Odd Roots
When dealing with odd roots, such as cube roots, negative numbers can yield real results. Specifically, the cube root of a negative number is also negative. This is in contrast to even roots, where the root of a negative number is not a real number. Thus, ∛-125 will yield a real number.
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Evaluation of Roots
To evaluate a root, you determine the number that satisfies the root equation. For ∛-125, you need to find a number that, when multiplied by itself three times, equals -125. In this case, the evaluation leads to -5, since (-5) × (-5) × (-5) = -125.
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