Write ∛64 using exponents and evaluate.

Verified step by step guidance

1. Recognize that the cube root of a number can be expressed as an exponent: \( \sqrt[3]{64} = 64^{1/3} \).
2. Identify that 64 is a power of 2: \( 64 = 2^6 \).
3. Substitute \( 64 \) with \( 2^6 \) in the expression: \( (2^6)^{1/3} \).
4. Apply the power of a power property of exponents: \( (a^m)^n = a^{m \cdot n} \).
5. Calculate the new exponent: \( 2^{6 \cdot (1/3)} = 2^2 \).

Key Concepts

Radicals and Exponents

Radicals are expressions that involve roots, such as square roots or cube roots. The cube root of a number 'a' is expressed as ∛a, which is equivalent to a raised to the power of 1/3. Understanding the relationship between radicals and exponents is crucial for manipulating and evaluating expressions involving roots.

Evaluating Cube Roots

To evaluate a cube root, you need to find a number that, when multiplied by itself three times, equals the original number. For example, ∛64 asks for a number that satisfies x³ = 64. Recognizing that 4 × 4 × 4 = 64 allows us to conclude that ∛64 = 4.

Properties of Exponents

Properties of exponents, such as the product of powers and power of a power, help simplify expressions involving exponents. When rewriting radicals as exponents, these properties can be applied to combine or simplify terms. For instance, knowing that a^(m/n) = ∛(a^m) can facilitate the evaluation of expressions like ∛64.