Problem 18 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$9^{x}=\frac{1}{\sqrt[3]{3}}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1/6\)

Step 1: Expressing the base

First, rewrite both sides of the equation using the same base. The number 9 is actually \(3^2\) and the number 1 over cube root of 3 is \((3^{-1/3})\). So, we rewrite the equation as \((3^2)^x = 3^{-1/3}\). This simplifies to \(3^{2x} = 3^{-1/3}\).

Step 2: Equating Exponents

Since the bases are the same on both sides of the equation, we can set the exponents equal to each other. Thus, \(2x = -1/3\).

Step 3: Solving for x

Next, solve the equation for x by dividing both sides by 2. Therefore, \(x = -1/6\).