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

Verified step by step guidance

1. Recognize that the expression \(\left( -\frac{64}{27} \right)^{\frac{1}{3}}\) represents the cube root of the fraction \(-\frac{64}{27}\).
2. 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: \(\left( \frac{a}{b} \right)^{\frac{1}{3}} = \frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}}\).
3. Apply this property to rewrite the expression as \(\frac{(-64)^{\frac{1}{3}}}{27^{\frac{1}{3}}}\).
4. Find the cube root of the numerator \(-64\). Since \(-64\) is a perfect cube, identify the number which when cubed gives \(-64\).
5. Find the cube root of the denominator \(27\). Since \(27\) is a perfect cube, identify the number which when cubed gives \(27\).

Key Concepts

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.