Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. - ∛5/8

Verified step by step guidance

1. Identify the expression given: the cube root of the fraction \( \frac{5}{8} \), written as \( \sqrt[3]{\frac{5}{8}} \).
2. Recall the property of radicals that allows you to separate the root of a fraction into the root of the numerator over the root of the denominator: \( \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \).
3. Apply this property to rewrite the expression as \( \frac{\sqrt[3]{5}}{\sqrt[3]{8}} \).
4. Recognize that \( \sqrt[3]{8} \) is a perfect cube since \( 8 = 2^3 \), so \( \sqrt[3]{8} = 2 \).
5. Simplify the expression to \( \frac{\sqrt[3]{5}}{2} \), which is the simplified form using the rules for radicals.

Key Concepts

Properties of Radicals

Radicals represent roots of numbers or expressions, such as square roots or cube roots. Understanding how to simplify and manipulate radicals using their properties, like the product and quotient rules, is essential for performing operations involving roots.

Cube Roots

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, ∛8 = 2 because 2³ = 8. Recognizing how to simplify cube roots and apply operations to them is key to solving problems involving ∛ expressions.

Quotient Rule for Radicals

The quotient rule states that the root of a quotient equals the quotient of the roots, i.e., ∛(a/b) = ∛a / ∛b, assuming all variables represent positive real numbers. This rule allows you to separate or combine radicals in division, facilitating simplification.