Sure! Let's solve the equation [tex]\( 32^{x+1} = 64 \)[/tex] step-by-step.
1. Rewrite the equation in terms of powers of 2:
- Notice that [tex]\( 32 \)[/tex] and [tex]\( 64 \)[/tex] can be expressed as powers of 2.
- Specifically, [tex]\( 32 = 2^5 \)[/tex] and [tex]\( 64 = 2^6 \)[/tex].
Therefore, we can rewrite the original equation as:
[tex]\[
(2^5)^{x+1} = 2^6
\][/tex]
2. Simplify the left side using the power of a power rule:
- According to the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[
2^{5(x+1)} = 2^6
\][/tex]
3. Since the bases on both sides of the equation are the same, we can equate the exponents:
- We now have:
[tex]\[
5(x+1) = 6
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- First, expand the left-hand side:
[tex]\[
5x + 5 = 6
\][/tex]
- Next, isolate [tex]\( x \)[/tex] by subtracting 5 from both sides:
[tex]\[
5x = 1
\][/tex]
- Finally, divide by 5:
[tex]\[
x = \frac{1}{5}
\][/tex]
Hence, the solution to the equation [tex]\( 32^{x+1} = 64 \)[/tex] is:
[tex]\[
\boxed{\frac{1}{5}}
\][/tex]
