To solve the equation [tex]\(32^{2c} = 8^{c+7}\)[/tex], we will follow these steps:
1. Rewrite the numbers as powers of 2:
- [tex]\(32\)[/tex] can be written as [tex]\(2^5\)[/tex].
- [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex].
So, the equation [tex]\(32^{2c} = 8^{c+7}\)[/tex] can be rewritten using these base-2 representations:
[tex]\[
(2^5)^{2c} = (2^3)^{c+7}
\][/tex]
2. Simplify the exponents:
Using the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[
2^{5 \cdot 2c} = 2^{3 \cdot (c+7)}
\][/tex]
Simplify the exponents:
[tex]\[
2^{10c} = 2^{3c + 21}
\][/tex]
3. Set the exponents equal to each other:
Since the bases are the same (both are 2), we can set the exponents equal to each other:
[tex]\[
10c = 3c + 21
\][/tex]
4. Solve for [tex]\(c\)[/tex]:
Isolate [tex]\(c\)[/tex] by subtracting [tex]\(3c\)[/tex] from both sides of the equation:
[tex]\[
10c - 3c = 21
\][/tex]
[tex]\[
7c = 21
\][/tex]
Divide both sides by 7:
[tex]\[
c = \frac{21}{7}
\][/tex]
[tex]\[
c = 3
\][/tex]
Thus, the value of [tex]\(c\)[/tex] is [tex]\(\boxed{3}\)[/tex].
