Evaluate each expression 125^1/3.

Verified step by step guidance

1. Recognize that the expression involves a negative fractional exponent: \(125^{\left(-\frac{1}{3}\right)}\).
2. Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\). Apply this to rewrite the expression as \(\frac{1}{125^{\frac{1}{3}}}\).
3. Understand that the fractional exponent \(\frac{1}{3}\) represents the cube root, so rewrite the denominator as \(\sqrt[3]{125}\).
4. Evaluate the cube root of 125 by finding the number which, when cubed, equals 125.
5. Write the final expression as \(\frac{1}{\sqrt[3]{125}}\) and simplify further if possible.

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a^(-n) equals 1 divided by a^n. This concept allows us to rewrite expressions with negative powers into fractions.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. Specifically, a^(m/n) means the n-th root of a raised to the m-th power, or equivalently, (a^(1/n))^m. This helps in simplifying expressions involving roots and powers.

Evaluating Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 125 is 5 because 5^3 = 125. Understanding cube roots is essential for simplifying expressions with fractional exponents like 1/3.