Evaluate each expression without using a calculator. 27^1/3

Verified step by step guidance

1. Recognize that the expression \$27^{1/3}$ represents the cube root of 27, because an exponent of \(\frac{1}{3}\) means the cube root.
2. Rewrite the expression as \(\sqrt[3]{27}\) to make it clearer that we are looking for a number which, when multiplied by itself three times, equals 27.
3. Recall that 27 is a perfect cube since \(3 \times 3 \times 3 = 27\).
4. Identify that the cube root of 27 is 3 because \$3^3 = 27$.
5. Therefore, \$27^{1/3}$ simplifies to 3.

Key Concepts

Exponents and Powers

Exponents represent repeated multiplication of a base number. For example, 27 can be written as 3^3 because 3 multiplied by itself three times equals 27. Understanding how to manipulate exponents is essential for simplifying expressions involving powers.

Fractional Exponents

A fractional exponent like 1/3 indicates a root; specifically, the denominator of the fraction is the root's degree. For instance, x^(1/3) means the cube root of x. This concept allows rewriting roots as exponents, facilitating easier evaluation and simplification.

Evaluating Cube Roots

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3^3 = 27. Recognizing perfect cubes helps in evaluating expressions like 27^(1/3) without a calculator.