Write each root using exponents and evaluate. ⁵√32

Verified step by step guidance

1. Recognize that the expression \( \sqrt[5]{32} \) is asking for the fifth root of 32.
2. Express the fifth root using exponents: \( 32^{1/5} \).
3. Recall that 32 can be rewritten as a power of 2: \( 32 = 2^5 \).
4. Substitute \( 32 = 2^5 \) into the expression: \( (2^5)^{1/5} \).
5. Apply the power of a power property \((a^m)^n = a^{m \cdot n}\) to simplify: \( 2^{5 \cdot (1/5)} \).

Key Concepts

Roots and Exponents

Roots and exponents are fundamental concepts in algebra that describe the relationship between numbers. The nth root of a number is the value that, when raised to the nth power, gives the original number. For example, the square root of 9 is 3 because 3² = 9. This relationship can also be expressed using fractional exponents, where the nth root of a number 'a' is represented as a^(1/n).

Evaluating Roots

Evaluating roots involves calculating the value of a number that satisfies the root equation. For instance, to evaluate the fifth root of 32 (⁵√32), we seek a number that, when raised to the power of 5, equals 32. This process often requires knowledge of perfect powers and can involve prime factorization to simplify the calculation.

Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors, which are the building blocks of all integers. For example, the prime factorization of 32 is 2^5, since 32 can be expressed as 2 multiplied by itself five times. Understanding prime factorization is crucial for simplifying roots and exponents, as it allows for easier evaluation of expressions involving powers.