In Exercises 47–54, find each cube root.________³√−27/1000

Verified step by step guidance

1. Identify the expression for which you need to find the cube root: \( \sqrt[3]{\frac{-27}{1000}} \).
2. Recognize 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 separately: \( \sqrt[3]{-27} \) and \( \sqrt[3]{1000} \).
4. Recall that \( -27 \) is \( -3^3 \) and \( 1000 \) is \( 10^3 \), so \( \sqrt[3]{-27} = -3 \) and \( \sqrt[3]{1000} = 10 \).
5. Combine the results to express the cube root of the fraction: \( \frac{-3}{10} \).

Key Concepts

Cube Roots

A 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 27 is 3, since 3 × 3 × 3 = 27. Cube roots can be positive or negative, as both will yield the same result when cubed.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. In the context of the given problem, -27/1000 is a rational number, and finding its cube root involves understanding how to handle fractions and negative values in root calculations.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with roots and powers. For instance, the cube root can be expressed using exponents as x^(1/3). This property allows for simplification and manipulation of expressions involving roots, especially when working with rational numbers.