if $32^{2c}=8^{c + 7}$, what is the value of $c$?\n1\n2\n3\n5

Explanation:

Step1: Rewrite bases as powers of 2

Since $32 = 2^5$ and $8=2^3$, we can rewrite the equation $32^{2c}=8^{c + 7}$ as $(2^5)^{2c}=(2^3)^{c + 7}$.

Step2: Apply power - of - a - power rule

The power - of - a - power rule $(a^m)^n=a^{mn}$. So, $(2^5)^{2c}=2^{5\times2c}=2^{10c}$ and $(2^3)^{c + 7}=2^{3\times(c + 7)}=2^{3c+21}$. Our equation becomes $2^{10c}=2^{3c + 21}$.

Step3: Set exponents equal

If $a^m=a^n$, then $m = n$ for $a>0,a\neq1$. So, $10c=3c + 21$.

Step4: Solve for c

Subtract $3c$ from both sides: $10c-3c=3c + 21-3c$, which gives $7c=21$. Then divide both sides by 7: $c=\frac{21}{7}=3$.

Answer:

C. 3