Certainly! Let's solve the equation step-by-step.
The given equation is:
[tex]\[
\sqrt{5^{x+2}} = 25^{x-2}
\][/tex]
### Step 1: Rewriting the Base
First, recognize that [tex]\(25\)[/tex] can be rewritten as a power of [tex]\(5\)[/tex]:
[tex]\[
25 = 5^2
\][/tex]
Hence, the equation becomes:
[tex]\[
\sqrt{5^{x+2}} = (5^2)^{x-2}
\][/tex]
### Step 2: Simplifying Exponents
Next, simplify the right-hand side using the power rule [tex]\( (a^m)^n = a^{mn} \)[/tex]:
[tex]\[
(5^2)^{x-2} = 5^{2(x-2)}
\][/tex]
So, the equation now looks like:
[tex]\[
\sqrt{5^{x+2}} = 5^{2(x-2)}
\][/tex]
### Step 3: Converting the Square Root
Let's convert the square root on the left-hand side to an exponent:
[tex]\[
\sqrt{5^{x+2}} = (5^{x+2})^{1/2}
\][/tex]
Using the power rule again, we get:
[tex]\[
(5^{x+2})^{1/2} = 5^{(x+2)/2}
\][/tex]
Now both sides of the equation have the same base:
[tex]\[
5^{(x+2)/2} = 5^{2(x-2)}
\][/tex]
### Step 4: Setting the Exponents Equal
Since the bases are the same, we can set the exponents equal to one another:
[tex]\[
\frac{x+2}{2} = 2(x-2)
\][/tex]
### Step 5: Solving the Equation
We now solve this linear equation:
[tex]\[
\frac{x+2}{2} = 2(x-2)
\][/tex]
First, clear the fraction by multiplying every term by 2:
[tex]\[
x + 2 = 4(x-2)
\][/tex]
Distribute the 4 on the right-hand side:
[tex]\[
x + 2 = 4x - 8
\][/tex]
Next, isolate [tex]\(x\)[/tex]. Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
2 = 3x - 8
\][/tex]
Add 8 to both sides:
[tex]\[
10 = 3x
\][/tex]
Finally, divide by 3:
[tex]\[
x = \frac{10}{3}
\][/tex]
### Solution
Thus, the solution to the equation [tex]\(\sqrt{5^{x+2}} = 25^{x-2}\)[/tex] is:
[tex]\[
x = \frac{10}{3}
\][/tex]
