Simplify each radical. Assume all variables represent positive real numbers. ⁸√5⁴

Verified step by step guidance

1. Recognize that the expression is an eighth root of 5 raised to the fourth power, which can be written as \(\sqrt[8]{5^{4}}\).
2. Recall the property of radicals that \(\sqrt[n]{a^{m}} = a^{\frac{m}{n}}\), so rewrite the expression as \(5^{\frac{4}{8}}\).
3. Simplify the fraction in the exponent: \(\frac{4}{8} = \frac{1}{2}\), so the expression becomes \(5^{\frac{1}{2}}\).
4. Recognize that \(5^{\frac{1}{2}}\) is equivalent to the square root of 5, written as \(\sqrt{5}\).
5. Therefore, the simplified form of the original radical is \(\sqrt{5}\).

Key Concepts

Radical Expressions and Roots

A radical expression involves roots such as square roots, cube roots, or higher-order roots. The nth root of a number is a value that, when raised to the nth power, gives the original number. For example, the 8th root of a number x is written as ⁿ√x or x^(1/8).

Properties of Exponents

Exponents follow specific rules that help simplify expressions, such as (a^m)^n = a^(m*n) and a^(m/n) = ⁿ√(a^m). These properties allow rewriting radicals as fractional exponents, making it easier to simplify expressions involving roots and powers.

Simplifying Radicals with Variables

When simplifying radicals with variables, assume variables represent positive real numbers to avoid complications with negative values. Simplification involves rewriting the expression using fractional exponents and reducing powers by dividing the exponent by the root index, ensuring the result is in simplest form.