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

Verified step by step guidance

Recognize that the expression $\left( -\frac{64}{27} \right)^{\frac{1}{3}}$ represents the cube root of the fraction $-\frac{64}{27}$.

Recall the property of exponents that allows you to take the cube root of a fraction by taking the cube root of the numerator and the denominator separately: $\left( \frac{a}{b} \right)^{\frac{1}{3}} = \frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}}$.

Apply this property to rewrite the expression as $\frac{(-64)^{\frac{1}{3}}}{27^{\frac{1}{3}}}$.

Find the cube root of the numerator $-64$. Since $-64$ is a perfect cube, identify the number which when cubed gives $-64$.

Find the cube root of the denominator $27$. Since $27$ is a perfect cube, identify the number which when cubed gives $27$.

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Key Concepts

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

Rational Exponents

Rational exponents represent roots and powers combined. An expression like a^(m/n) means the nth root of a raised to the mth power. For example, a^(1/3) is the cube root of a, and a^(2/3) is the cube root of a squared.

Cube Roots of Negative Numbers

The cube root of a negative number is also negative because cubing a negative number results in a negative number. For instance, the cube root of -64 is -4 since (-4)^3 = -64. This differs from even roots, which are not defined for negative numbers in real numbers.

Simplifying Fractional Expressions

When dealing with fractional bases like (-64/27), apply the root to both numerator and denominator separately. For example, the cube root of (-64/27) equals the cube root of -64 divided by the cube root of 27, simplifying the expression step-by-step.

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