If begin mathsize 18px style a equals end style14, then the solution of the exponential equation open parentheses italic 25 over italic 9 close parentheses to the power of x minus a end exponent italic equals italic 81 over italic 625  is:





Given the exponential equation:

\[
\left(\frac{25}{9}\right)^{x-14} = \frac{81}{625}
\]

Let's solve this step-by-step.

### Step 1: Rewrite the bases in terms of prime factors

Express \(25\), \(9\), \(81\), and \(625\) using their prime factorizations:

\[
25 = 5^2, \quad 9 = 3^2, \quad 81 = 3^4, \quad 625 = 5^4
\]

Substitute these into the equation:

\[
\left(\frac{5^2}{3^2}\right)^{x-14} = \frac{3^4}{5^4}
\]

### Step 2: Simplify the equation

Rewrite the left side:

\[
\left(\frac{5^2}{3^2}\right)^{x-14} = \left(\frac{5}{3}\right)^{2(x-14)}
\]

And the right side:

\[
\frac{3^4}{5^4} = \left(\frac{3}{5}\right)^4 = \left(\frac{5}{3}\right)^{-4}
\]

So the equation now becomes:

\[
\left(\frac{5}{3}\right)^{2(x-14)} = \left(\frac{5}{3}\right)^{-4}
\]

### Step 3: Set the exponents equal

Since the bases are the same, set the exponents equal to each other:

\[
2(x-14) = -4
\]

### Step 4: Solve for \(x\)

\[
2x - 28 = -4
\]

Add 28 to both sides:

\[
2x = 24
\]

Divide by 2:

\[
x = 12
\]

### Solution

The solution to the exponential equation is:

\[
x = 12
\]

Would you like a detailed explanation or have any questions?

### Related Questions:

1. How do you solve exponential equations with different bases?
2. What is the process for simplifying expressions involving exponents?
3. How do you convert between fractional and negative exponents?
4. What are the properties of exponents used in solving exponential equations?
5. How do you apply the laws of exponents to equations with prime factorization?

### Tip:
When dealing with exponential equations, it's often helpful to express all terms with the same base, which allows you to set the exponents equal to each other and solve for the unknown.

If a = 14, then the solution of the exponential equation (25/9)^(x - a) = 81/625 is:

This page provides a detailed step-by-step solution for solving the exponential equation (25/9)^(x-14) = 81/625. Learn how to express terms with the same base and solve for the unknown exponent using exponent rules. Suitable for students in Grades 10-12.

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