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

Verified step by step guidance

1. Identify the expression to evaluate: $(-\frac{64}{27}{)}^{\frac{1}{3}}$.
2. Recognize that raising a number to the power of $\frac{1}{3}$ is equivalent to taking the cube root of that number.
3. Rewrite the expression as the cube root: $\sqrt[3]{-\frac{64}{27}}$.
4. Apply the property of cube roots: $\sqrt[3]{\frac{a}{b}}=\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$.
5. Calculate the cube root of the numerator and the denominator separately: $\sqrt[3]{-64}$ and $\sqrt[3]{27}$.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers in a compact form. For example, an exponent of 1/3 indicates the cube root of a number. This concept allows us to express operations involving roots using exponentiation, making calculations more straightforward.

Negative Numbers and Roots

When dealing with negative numbers, the cube root can yield real results, unlike even roots. The cube root of a negative number is also negative, which is essential for evaluating expressions like (-64/27)^(1/3). Understanding how roots behave with negative values is crucial for accurate calculations.

Fractional Bases

Evaluating expressions with fractional bases involves simplifying the fraction before applying the exponent. In this case, -64/27 can be simplified to its cube root by separately finding the cube roots of the numerator and denominator. This concept is vital for breaking down complex expressions into manageable parts.