Problem 18 - Introductory and Intermediate Algebra for College Students 4th

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 \(9^{x}=\frac{1}{\sqrt[3]{3}}\) is \(x=-1/6\).

Step 1: Express Each Side as a Power of the Same Base

Express both the left and right side of the equation as powers of the base number 3. The base number 3 was chosen because it is the common base for both 9 and 1/3. So, \(9^{x}\) can be written as \((3^{2})^{x}\) and \(\frac{1}{\sqrt[3]{3}}\) can be written as \(3^{-1/3}\)

Step 2: Using Laws of Exponents Simplify the Bases

Simplify the left side of the equation using the property of exponents that says \((a^{b})^{c} = a^{bc}\). So, the left side of the equation simplifies to \(3^{2x}\). Hence, our equation is now \(3^{2x}=3^{-1/3}\)

Step 3: Equate the Exponents

Since we have expressed both sides of the equation as powers of the same base (3), we can equate the exponents to each other. Therefore, setting the exponents equal to each other gives us \(2x=-1/3\)

Step 4: Solve for \(x\)

To isolate \(x\), divide both sides of the equation by 2: \(x=-1/6\)