(1/9)^x+4=729

2 min read Jun 16, 2024

Solving the Exponential Equation: (1/9)^x+4 = 729

This article will guide you through the process of solving the exponential equation (1/9)^x+4 = 729.

The equation involves an exponential expression with an unknown exponent, x. Our goal is to find the value of x that makes the equation true.

Express both sides of the equation with the same base:
- Recognize that 729 is 9 cubed (9^3).
- Rewrite (1/9) as 9 raised to the power of -1 (9^-1).
- The equation now becomes: (9^-1)^x+4 = 9^3
Simplify using exponent rules:
- Apply the rule: (a^m)^n = a^(m*n)
- The equation becomes: 9^(-x-4) = 9^3
Equate the exponents:
- Since the bases are now the same, we can equate the exponents: -x - 4 = 3
Solve for x:
- Add 4 to both sides: -x = 7
- Multiply both sides by -1: x = -7

Therefore, the solution to the equation (1/9)^x+4 = 729 is x = -7.

To verify our answer, we can substitute x = -7 back into the original equation:

(1/9)^(-7)+4 = (1/9)^(-3) = 9^3 = 729

This confirms that our solution is correct.