Find each root. See Example 3. ∛0.125

Verified step by step guidance

1. Recognize that the problem asks for the cube root of 0.125, which means finding a number \( x \) such that \( x^3 = 0.125 \).
2. Express 0.125 as a fraction to make it easier to work with: \( 0.125 = \frac{1}{8} \).
3. Rewrite the cube root expression using fractional exponents: \( \sqrt[3]{0.125} = \left( \frac{1}{8} \right)^{\frac{1}{3}} \).
4. Apply the property of exponents to the fraction: \( \left( \frac{1}{8} \right)^{\frac{1}{3}} = \frac{1^{\frac{1}{3}}}{8^{\frac{1}{3}}} \).
5. Calculate the cube root of numerator and denominator separately: \( 1^{\frac{1}{3}} = 1 \) and \( 8^{\frac{1}{3}} = 2 \), so the cube root is \( \frac{1}{2} \).

Key Concepts

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 8 is 2 because 2 × 2 × 2 = 8. It is denoted as ∛x.

Fractional Exponents

Fractional exponents represent roots; specifically, an exponent of 1/n corresponds to the nth root of a number. For example, x^(1/3) is the cube root of x. This notation helps in simplifying and calculating roots using exponent rules.

Decimal to Fraction Conversion

Converting decimals to fractions can simplify root calculations. For instance, 0.125 equals 1/8, making it easier to find roots since roots of fractions can be found by taking roots of numerator and denominator separately.