In Exercises 83–90, 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.
2. Recall that finding the cube root of a number is the same as finding a number that, when multiplied by itself three times, equals the original number.
3. Consider the number 3, since $3\times 3\times 3=27$.
4. Verify that $3^3=27$, confirming that 3 is indeed the cube root of 27.
5. Conclude that ${27}^{1/3}=3$.

Key Concepts

Exponents and Roots

Exponents represent repeated multiplication, while roots are the inverse operation. For example, the expression 27^(1/3) indicates the cube root of 27, which asks for a number that, when multiplied by itself three times, equals 27. Understanding how to manipulate exponents and roots is essential for evaluating expressions like this.

Cube Roots

The cube root of a number x, denoted as x^(1/3), is the value that, when cubed, gives x. In this case, finding 27^(1/3) means identifying a number that, when raised to the power of three, results in 27. Recognizing perfect cubes, such as 1, 8, and 27, helps in quickly determining cube roots.

Perfect Cubes

Perfect cubes are numbers that can be expressed as the cube of an integer. For instance, 1 (1^3), 8 (2^3), and 27 (3^3) are perfect cubes. Knowing these values allows for easier evaluation of cube roots, as one can directly identify that the cube root of 27 is 3, since 3 multiplied by itself three times equals 27.