Problem 82 - College Algebra

Question

In Exercises \(77-84,\) evaluate each expression without using a calculator. $$8^{2 / 3}$$

Step-by-Step Solution

Answer

The evaluation of the expression \(8^{2 / 3}\) without using a calculator is 4.

Step 1: Break Down the Exponent

Before we proceed we need to know that the exponent \(2/3\) tells us to find the cube root of 8 first \(3\) and then square it \(2\). So first break down the exponent into its constituents - first the cube root, followed by the square. We write \(8^{2/3}\) as \((8^{1/3})^2\)

Step 2: Evaluate the Cube Root

Now we find the cube root of 8. The cube root of a number is a value that, when cubed, gives the original number. Based on our knowledge of numbers, we know that \(2*2*2 = 8\). Therefore, the cube root of 8 is 2. So the expression now simplifies to \(2^2\).

Step 3: Evaluate the Square

Now, we square 2 (raise 2 to the second power). Squaring a number is simply multiplying that number by itself. Therefore, \(2^2 = 2*2 = 4\).