Question

Evaluate each expression. $$(-125)^{-4 / 3}$$

   Evaluate each expression.
$$(-125)^{-4 / 3}$$

A Graphical Approach to Precalculus with Limits

A Graphical Approach to Precalculus with Limits

John Hornsby,… 7th Edition

Chapter 5, Problem 10

Instant Answer

Solved by Expert Nick Johnson

08/14/2020

Step 1

So, $(-125)^{-4 / 3}$ is equivalent to $\frac{1}{(-125)^{4 / 3}}$. Show more…

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Evaluate each expression. $$(-125)^{-4 / 3}$$

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Key Concepts

Rational Exponents

Rational exponents express both a power and a root in one notation. When an exponent is written as a fraction (for example, a^(m/n)), the denominator indicates the nth root of the base and the numerator indicates the power to which the result is raised. This concept allows for the evaluation of roots and powers in a single step, facilitating the manipulation and simplification of complex expressions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. In other words, for any nonzero number a, a^(-n) is equivalent to 1/(a^n). This concept is critical for rewriting expressions such that all exponents become positive, which simplifies further evaluation and understanding.

Cube Roots of Negative Numbers

When dealing with cube roots, the odd index allows for the extraction of the root of a negative number while staying within the real number system. This means that the cube root of a negative number will yield a negative result. Understanding this property is essential when evaluating expressions that involve both negative bases and rational exponents with an odd denominator.

Even Exponents

Even exponents, when applied to a negative base, result in a positive outcome because the product of an even number of negative factors is positive. This is an important principle when raising numbers derived from roots, ensuring that the final result's sign is correctly determined after applying the exponent.

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