Write each root using exponents and evaluate. ∛216

Verified step by step guidance

1. Recognize that the cube root of a number is the same as raising that number to the power of \( \frac{1}{3} \).
2. Express the cube root of 216 as \( 216^{\frac{1}{3}} \).
3. Factor 216 into its prime factors: \( 216 = 2^3 \times 3^3 \).
4. Apply the exponent rule \( (a^m \times b^n)^p = a^{m \cdot p} \times b^{n \cdot p} \) to \( (2^3 \times 3^3)^{\frac{1}{3}} \).
5. Simplify the expression to find \( 2^{3 \cdot \frac{1}{3}} \times 3^{3 \cdot \frac{1}{3}} \).

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 cube root (∛) of a number x is expressed as x^(1/3), indicating that x is raised to the power of one-third.

Evaluating Roots

Evaluating roots involves finding the numerical value that satisfies the root equation. For instance, to evaluate ∛216, we seek a number that, when multiplied by itself three times, equals 216. This process often requires knowledge of perfect cubes and can be simplified by recognizing that 6 × 6 × 6 = 216, thus ∛216 = 6.

Perfect Cubes

Perfect cubes are numbers that can be expressed as the cube of an integer. Understanding perfect cubes is essential for evaluating cube roots efficiently. For example, the perfect cubes up to 10 are 1 (1^3), 8 (2^3), 27 (3^3), 64 (4^3), 125 (5^3), and 216 (6^3). Recognizing these values helps in quickly determining the cube roots of numbers.