To solve the equation [tex]\(6^{2x + 2} \cdot 6^{3x} = 1\)[/tex], we can follow these steps:
1. Combine the exponents:
The equation can be simplified by using the properties of exponents. When multiplying like bases, you add the exponents:
[tex]\[
6^{2x + 2} \cdot 6^{3x} = 6^{(2x + 2) + (3x)} = 6^{5x + 2}
\][/tex]
2. Set the simplified equation equal to 1:
[tex]\[
6^{5x + 2} = 1
\][/tex]
3. Recognize a property of exponents:
We know that any nonzero number to the power of 0 is 1. Therefore, for [tex]\(6^{5x + 2}\)[/tex] to equal 1, the exponent must be 0:
[tex]\[
5x + 2 = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
5x + 2 = 0
\][/tex]
[tex]\[
5x = -2
\][/tex]
[tex]\[
x = \frac{-2}{5}
\][/tex]
5. Conclude the solution:
[tex]\[
x = -0.4
\][/tex]
Thus, the solution to the equation [tex]\(6^{2x + 2} \cdot 6^{3x} = 1\)[/tex] is:
[tex]\[
x = -0.4
\][/tex]
