Problem 133 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$

Step-by-Step Solution

Answer

The expression \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 49\) is false. The corrected expression should be \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 7\).

Step 1: Understand the Fractional Exponent

Firstly, we need to know that a fractional exponent indicates a root. In this case, an exponent of \(\frac{1}{2}\) represents a square root. So, \(7^{\frac{1}{2}}\) is the square root of 7.

Step 2: Apply the Exponential Rule

By applying the rule \(a^m \cdot a^n = a^{m+n}\), we get \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 7^{1} = 7\). So, the statement is false.

Step 3: Correct the Statement

To make the expression true, the statement should be \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 7\), because the square root of 7 multiplied by the square root of 7 equals to 7.