((5)/(3))^(x+1)*((9)/(25))^(x^(2)+2x-11)=((5)/(3))^(9)

Jasonbradley

3 min read

·

Jun 16, 2024

54

Share

Solving Exponential Equations: A Step-by-Step Guide

This article will guide you through solving the exponential equation:

(5/3)^(x+1) * (9/25)^(x^2 + 2x - 11) = (5/3)^9

Let's break down the solution step-by-step:

1. Express All Bases as the Same Value

Our goal is to have all terms with the same base. Notice that (9/25) can be expressed as (3/5)^2. Substituting this into the equation gives us:

(5/3)^(x+1) * ( (3/5)^2 )^(x^2 + 2x - 11) = (5/3)^9

2. Simplify Using Power of a Power Rule

We can simplify the left-hand side using the power of a power rule: (a^m)^n = a^(m*n). Applying this:

(5/3)^(x+1) * (3/5)^(2x^2 + 4x - 22) = (5/3)^9

3. Express All Terms with the Same Base

To further simplify, let's express (3/5) as (5/3)^(-1):

(5/3)^(x+1) * ( (5/3)^(-1) )^(2x^2 + 4x - 22) = (5/3)^9

Applying the power of a power rule again:

(5/3)^(x+1) * (5/3)^(-2x^2 - 4x + 22) = (5/3)^9

4. Combine Exponents

Now, we can combine the terms on the left-hand side using the rule: a^m * a^n = a^(m+n):

(5/3)^(x+1 - 2x^2 - 4x + 22) = (5/3)^9

Simplifying the exponent:

(5/3)^(-2x^2 - 3x + 23) = (5/3)^9

5. Solve for x

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

-2x^2 - 3x + 23 = 9

Rearranging the equation to form a quadratic:

2x^2 + 3x - 14 = 0

Factoring the quadratic equation:

(2x - 7)(x + 2) = 0

Therefore, the solutions for x are:

x = 7/2 or x = -2

Conclusion

By applying the rules of exponents and simplifying the equation, we successfully solved for x. The solutions to the equation are x = 7/2 and x = -2.