Problem 88 - College Algebra

Question

Evaluate each expression without using a calculator. $$8^{\frac{2}{3}}$$

Step-by-Step Solution

Answer

The evaluation of the expression \(8^{\frac{2}{3}}\) gives us the result 4.

Step 1: Break the Exponential Fraction into Root and Power

First, rewrite the exponent as a radical expression. In \(8^{\frac{2}{3}}\), the denominator is 3, which corresponds to the cube root, and the numerator is 2, which represents squaring. So, this can be rewritten as \(\sqrt[3]{8^2}\).

Step 2: Calculate the Power

Next, calculate the power i.e., square 8. This gives us 64. Substitute it back into our expression: \(\sqrt[3]{64}\).

Step 3: Evaluate the Root

Finally, evaluate the cube root of 64. The cube root of any number y is the number x such that when x is cubed, it gives y. So, in this case, the cube root of 64 can be found by identifying a number that, when cubed, equals 64. And this number is 4. Therefore, the cube root of 64 is 4.