Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. ∜2/25

Verified step by step guidance

1. Recognize that the expression involves a fourth root (also called the fourth root or the 1/4 power) of a fraction: \(\sqrt[4]{\frac{2}{25}}\).
2. Use the property of radicals that allows you to separate the root of a fraction into the root of the numerator divided by the root of the denominator: \(\frac{\sqrt[4]{2}}{\sqrt[4]{25}}\).
3. Simplify the denominator by recognizing that \(25\) is a perfect square, and since we are taking the fourth root, express \(25\) as \$5^2$ and then apply the fourth root: \(\sqrt[4]{25} = \sqrt[4]{5^2}\).
4. Rewrite the fourth root of \$5^2$ as an exponent: \(5^{\frac{2}{4}} = 5^{\frac{1}{2}}\), which is the square root of 5.
5. Express the simplified form of the original expression as \(\frac{\sqrt[4]{2}}{5^{\frac{1}{2}}}\) or \(\frac{\sqrt[4]{2}}{\sqrt{5}}\), which is the simplified radical form.

Key Concepts

Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, and fourth roots (∜). The fourth root of a number is the value that, when raised to the fourth power, equals the original number. Understanding how to interpret and manipulate these roots is essential for simplifying expressions like ∜(2/25).

Properties of Exponents and Radicals

Radicals can be rewritten using fractional exponents, where the nth root of a number is expressed as that number raised to the 1/n power. For example, ∜(2/25) can be written as (2/25)^(1/4). This property allows the use of exponent rules to simplify or manipulate expressions more easily.

Simplification of Radical Expressions

Simplifying radicals often involves factoring the radicand into perfect powers and reducing the expression accordingly. Since variables are positive, we can safely apply root and exponent rules without considering absolute values. For fractions, the root applies to numerator and denominator separately, enabling simplification of each part.