(2/3)^x*(9/8)^x=27/64

2 min read Jun 16, 2024

Solving the Equation (2/3)^x * (9/8)^x = 27/64

This article will guide you through solving the equation (2/3)^x * (9/8)^x = 27/64. We will use the properties of exponents and simplify the equation to find the value of x.

1. Simplifying the Equation

First, we need to simplify the left side of the equation. Notice that both terms have the same base raised to the power of x.

(2/3)^x * (9/8)^x = [(2/3) * (9/8)]^x

We can multiply the fractions inside the brackets:

[(2/3) * (9/8)]^x = (3/4)^x

Now our equation looks like this:

(3/4)^x = 27/64

2. Expressing Both Sides with the Same Base

The next step is to express both sides of the equation with the same base. We can rewrite 27/64 and 3/4 as powers of the same base:

27/64 = (3/4)^3
3/4 = (3/4)^1

Substituting these back into the equation, we get:

(3/4)^x = (3/4)^3

3. Solving for x

Since the bases are the same, we can equate the exponents:

x = 3

Conclusion

Therefore, the solution to the equation (2/3)^x * (9/8)^x = 27/64 is x = 3.