Certainly! Let's solve the equation [tex]\( 3^{2x + 1} = 9^{2x - 1} \)[/tex] step by step.
### Step 1: Rewrite the equation using the same base
Notice that [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex]. So we can rewrite the right-hand side of the equation using the same base of 3:
[tex]\[ 9^{2x - 1} = (3^2)^{2x - 1} \][/tex]
### Step 2: Apply the power rule
Using the power rule [tex]\((a^m)^n = a^{mn}\)[/tex], the right-hand side becomes:
[tex]\[ (3^2)^{2x - 1} = 3^{2(2x - 1)} \][/tex]
### Step 3: Simplify the exponent
Now simplify the exponent on the right-hand side:
[tex]\[ 3^{2(2x - 1)} = 3^{4x - 2} \][/tex]
### Step 4: Set the exponents equal
Since we now have the same base on both sides of the equation, we can set the exponents equal to each other:
[tex]\[ 2x + 1 = 4x - 2 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Let's solve this equation for [tex]\( x \)[/tex]:
1. Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ 1 = 2x - 2 \][/tex]
2. Add 2 to both sides:
[tex]\[ 3 = 2x \][/tex]
3. Divide both sides by 2:
[tex]\[ x = \frac{3}{2} \][/tex]
### Conclusion
The solution to the equation [tex]\( 3^{2x + 1} = 9^{2x - 1} \)[/tex] is:
[tex]\[ x = \frac{3}{2} \][/tex]
And that's the detailed step-by-step solution!
