Problem 15 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$6^{\frac{x-3}{4}}=\sqrt{6}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\)

Step 1: Express the right-hand side to a power of the same base

To compare the exponents on each side of the equation, we need both sides to have the same base. Since \(6^{\frac{x-3}{4}} = \sqrt{6}\), we first need to express \(\sqrt{6}\) as a power. Recall that the square root can be expressed as a half power, so we can rewrite \(\sqrt{6}\) as \(6^{\frac{1}{2}}\). Therefore, the equation becomes \(6^{\frac{x-3}{4}} = 6^{\frac{1}{2}}\).

Step 2: Equate the exponents

Now that both sides are expressed to the base of 6, the given equation is in the form where the base of the powers is the same. Hence, we can equate the exponents as \(\frac{x-3}{4} = \frac{1}{2}\)

Step 3: Solve for the unknown variable

To isolate \(x\), we first multiply both sides of the equation by 4, canceling out the denominator in the left-hand side: \(x-3 = 2\). Adding 3 to both sides leads to \(x = 5\)