Certainly! Let's solve the equation step-by-step:
Given the equation:
[tex]\[
5^{x^2 + 2x + 7} = 125^{2x + 1}
\][/tex]
### Step 1: Simplify the Bases
First, we know that [tex]\(125\)[/tex] can be expressed as a power of [tex]\(5\)[/tex]:
[tex]\[
125 = 5^3
\][/tex]
Thus, we can rewrite the given equation as:
[tex]\[
5^{x^2 + 2x + 7} = (5^3)^{2x + 1}
\][/tex]
### Step 2: Simplify the Exponents
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we can rewrite the right side:
[tex]\[
(5^3)^{2x + 1} = 5^{3(2x + 1)} = 5^{6x + 3}
\][/tex]
Now the equation becomes:
[tex]\[
5^{x^2 + 2x + 7} = 5^{6x + 3}
\][/tex]
### Step 3: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
x^2 + 2x + 7 = 6x + 3
\][/tex]
### Step 4: Rearrange and Simplify the Equation
Rearrange the equation to bring all the terms to one side:
[tex]\[
x^2 + 2x + 7 - 6x - 3 = 0
\][/tex]
Combine like terms:
[tex]\[
x^2 - 4x + 4 = 0
\][/tex]
### Step 5: Solve the Quadratic Equation
Now we have a quadratic equation:
[tex]\[
x^2 - 4x + 4 = 0
\][/tex]
We can solve this quadratic equation using various methods such as factoring, completing the square, or using the quadratic formula. In this case, the equation factors nicely:
[tex]\[
(x - 2)^2 = 0
\][/tex]
So, we find:
[tex]\[
x - 2 = 0
\][/tex]
Thus:
[tex]\[
x = 2
\][/tex]
### Conclusion
By solving the equation, we find:
[tex]\[
x = 2
\][/tex]
Therefore, the solution to the equation [tex]\(5^{x^2 + 2x + 7} = 125^{2x + 1}\)[/tex] is:
[tex]\[
x = 2
\][/tex]
