Question

if $32^{2c}=8^{c + 7}$, what is the value of $c$?
5
1
2
3

Explanation:

Step1: Express bases as powers of 2

Since \(32 = 2^5\) and \(8 = 2^3\), rewrite the equation:
\((2^5)^{2c} = (2^3)^{c + 7}\)

Step2: Apply exponent rules

Using \((a^m)^n = a^{mn}\), we get:
\(2^{10c} = 2^{3(c + 7)}\)

Step3: Set exponents equal (since bases are equal)

\(10c = 3(c + 7)\)

Step4: Solve for \(c\)

Expand the right side: \(10c = 3c + 21\)
Subtract \(3c\) from both sides: \(7c = 21\)
Divide by 7: \(c = 3\)

Answer:

3