In Exercises 47–54, find each cube root._____³√1/125

Verified step by step guidance

1. Identify the expression as a cube root: \( \sqrt[3]{\frac{1}{125}} \).
2. Recall that the cube root of a fraction \( \frac{a}{b} \) is \( \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \).
3. Apply the cube root to both the numerator and the denominator: \( \frac{\sqrt[3]{1}}{\sqrt[3]{125}} \).
4. Recognize that \( \sqrt[3]{1} = 1 \) because any number to the power of 0 is 1.
5. Determine \( \sqrt[3]{125} \) by finding a number that, when multiplied by itself three times, equals 125.

Key Concepts

Cube Roots

A cube root of a number 'x' is a value 'y' such that y³ = x. This means that when 'y' is multiplied by itself three times, it equals 'x'. For example, the cube root of 8 is 2, since 2³ = 8. Understanding cube roots is essential for solving problems involving cubic equations or expressions.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In the case of 1/125, both 1 and 125 are integers, making it a rational number. Recognizing rational numbers is important when dealing with roots, as they can often simplify calculations.

Properties of Exponents

The properties of exponents govern how to manipulate expressions involving powers. For cube roots, the property states that (a/b)^(1/3) = a^(1/3) / b^(1/3). This property allows us to separate the numerator and denominator when finding roots, making it easier to compute the cube root of fractions like 1/125.