Let's solve the equation [tex]\(6^{2x + 2} \cdot 6^{3x} = 1\)[/tex] step-by-step.
1. Combine Exponents:
Given the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we can combine the exponents on the left-hand side.
[tex]\[
6^{(2x + 2)} \cdot 6^{3x} = 6^{(2x + 2 + 3x)} = 6^{(5x + 2)}
\][/tex]
So the equation now looks like:
[tex]\[
6^{5x + 2} = 1
\][/tex]
2. Interpret the Exponential Equation:
Recall that any non-zero number raised to the power of 0 is 1. Hence, we need the exponent on the left to be zero for the entire term to equal 1.
[tex]\[
6^{0} = 1
\][/tex]
Therefore, we set the exponent equal to 0:
[tex]\[
5x + 2 = 0
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Subtract 2 from both sides:
[tex]\[
5x = -2
\][/tex]
Divide both sides by 5:
[tex]\[
x = \frac{-2}{5}
\][/tex]
Hence,
[tex]\[
x = -0.4
\][/tex]
So the solution to the equation [tex]\(6^{2x + 2} \cdot 6^{3x} = 1\)[/tex] is
[tex]\[
x = -0.4
\][/tex]
