Question

Solve the following exponential equation: 25^3x+1=5^x-6



A
x=72x=\frac{7}{2}x=
2
7
​


B
x=85x=\frac{8}{5}x=
5
8
​


C
x=−72x=-\frac{7}{2}x=−
2
7
​


D
x=−85x=-\frac{8}{5}x=−
5
8
​

Answer 1

To solve the equation \( 25^{3x+1} = 5^{x-6} \), let's first express everything in terms of the same base.

Since \( 25 \) can be written as \( 5^2 \), we have:

\[
25^{3x+1} = (5^2)^{3x+1} = 5^{2(3x+1)} = 5^{6x + 2}
\]

Now substituting back into the original equation gives us:

\[
5^{6x + 2} = 5^{x - 6}
\]

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

\[
6x + 2 = x - 6
\]

Now, we will solve for \( x \):

1. Subtract \( x \) from both sides:

\[
6x - x + 2 = -6
\]

This simplifies to:

\[
5x + 2 = -6
\]

2. Next, subtract \( 2 \) from both sides:

\[
5x = -6 - 2
\]

This results in:

\[
5x = -8
\]

3. Now, divide by \( 5 \):

\[
x = -\frac{8}{5}
\]

Thus, the solution is:

\[
x = -\frac{8}{5}
\]

The answer corresponds to option D:

**D \(x = -\frac{8}{5}\)**.