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

Verified step by step guidance

1. Recognize that the expression \( \sqrt[8]{5^4} \) is asking for the eighth root of \( 5^4 \).
2. Recall the property of radicals and exponents: \( \sqrt[n]{a^m} = a^{m/n} \).
3. Apply this property to the expression: \( \sqrt[8]{5^4} = 5^{4/8} \).
4. Simplify the fraction \( \frac{4}{8} \) to its simplest form.
5. Rewrite the expression using the simplified exponent: \( 5^{1/2} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. The notation ⁸√ indicates the eighth root of a number. Understanding how to manipulate these expressions is crucial for simplification, especially when dealing with exponents and variables.

Exponents and Roots

Exponents represent repeated multiplication, while roots are the inverse operation. For example, the expression ⁸√x can be rewritten as x^(1/8). This relationship allows us to convert between radical and exponential forms, which is essential for simplifying radical expressions.

Properties of Exponents

The properties of exponents, such as the product of powers and power of a power, are fundamental in simplifying expressions. For instance, when simplifying ⁸√5⁴, we can apply the property that states a^(m/n) = a^(m)^(1/n) to rewrite the expression in a more manageable form.