\left(\frac{9}{\sqrt{3}}\right)^{2 x}=1 / 81
Answer (1)
Simplify the base: 3 9 = 3 3/2 .
Rewrite the right side: 81 1 = 3 − 4 .
Equate the exponents: 3 x = − 4 .
Solve for x : x = − 3 4 .
Explanation
Problem Analysis We are given the equation ( 3 9 ) 2 x = 1/81 and we need to solve for x .
Simplifying the Base First, let's simplify the base on the left side. We have 3 9 = 3 1/2 3 2 = 3 2 − 1/2 = 3 3/2 .
Rewriting the Right Side Next, let's rewrite the right side as a power of 3. We have 81 1 = 3 4 1 = 3 − 4 .
Rewriting the Equation Now, the equation becomes ( 3 3/2 ) 2 x = 3 − 4 .
Simplifying the Left Side Simplifying the left side, we get 3 3 x = 3 − 4 .
Equating the Exponents Since the bases are equal, we can equate the exponents: 3 x = − 4 .
Solving for x Finally, solving for x , we get x = − 3 4 .
Examples
Imagine you're adjusting the settings on a sound equalizer. The equation we solved is similar to how you might calculate the precise adjustments needed to amplify or reduce certain frequencies. By understanding exponential relationships, you can fine-tune audio settings, control signal strengths in communication systems, or even model population growth in biology. This type of problem provides a foundation for understanding how different parameters interact and influence outcomes in various real-world scenarios.
