Question: Solve the following elementary exponential equation. \[ \left(\frac{1}{3}\right)^{3x+5} =…

Solve the following elementary exponential equation.

\[ \left(\frac{1}{3}\right)^{3x+5} = \left(\frac{1}{9}\right)^3 \]

Enter your answer (opens in new window)

Keyboard

\( x = \boxed{\phantom{0}} \)

Solution

To solve the exponential equation \[ \left(\frac{1}{3}\right)^{3x+5} = \left(\frac{1}{9}\right)^3 \] we’ll follow these steps: First, express \( \frac{1}{9} \) as a power of \( \frac{1}{3} \): \[ \frac{1}{9} = \left(\frac{1}{3}\right)^2 \] Now substitute this into the equation: \[ \left(\frac{1}{3}\right)^{3x+5} = \left(\left(\frac{1}{3}\right)^2\right)^3 \] Simplify the right side by using the power of a power property \((a^m)^n = a^{m \cdot n}\): \[ \left(\left(\frac{1}{3}\right)^2\right)^3 = \left(\frac{1}{3}\right)^{2 \cdot 3} = \left(\frac{1}{3}\right)^6 \] Now both sides of the equation have the same base: \[ \left(\frac{1}{3}\right)^{3x+5} = \left(\frac{1}{3}\right)^6 \] Since the bases are the same, we can set the exponents equal to each other: \[ 3x + 5 = 6 \] Subtract 5 from both sides: \[ 3x = 1 \] Divide both sides by 3: \[ x = \frac{1}{3} \] The solution is \( x = \frac{1}{3} \).