Write each root using exponents and evaluate. ⁵√32

Verified step by step guidance

1. Recognize that the expression ⁵√32 represents the fifth root of 32, which can be rewritten using exponents as \$32^{\frac{1}{5}}$.
2. Express 32 as a power of a prime number. Since 32 is \$2^5\(, rewrite the expression as \)\left(2^5\right)^{\frac{1}{5}}$.
3. Apply the power of a power property of exponents: \(\left(a^m\right)^n = a^{m \times n}\). So, \(\left(2^5\right)^{\frac{1}{5}} = 2^{5 \times \frac{1}{5}}\).
4. Simplify the exponent multiplication: \$5 \times \frac{1}{5} = 1\(, so the expression becomes \)2^1$.
5. Interpret the result: \$2^1$ is simply 2, which is the value of the fifth root of 32.

Key Concepts

Radicals and Roots

A radical expression involves roots, such as square roots or fifth roots, which represent the inverse operation of exponentiation. The fifth root of a number is the value that, when raised to the power of 5, equals the original number. Understanding how to interpret and manipulate roots is essential for solving the problem.

Exponential Notation for Roots

Roots can be expressed using fractional exponents, where the nth root of a number is written as that number raised to the power of 1/n. For example, the fifth root of 32 can be written as 32^(1/5). This notation allows the use of exponent rules to simplify and evaluate roots.

Evaluating Powers and Simplification

Evaluating expressions with exponents involves applying the rules of exponents and recognizing powers of numbers. Since 32 is 2 raised to the 5th power, rewriting 32 as 2^5 and then applying the fractional exponent simplifies the evaluation, making it easier to find the numerical value of the root.