Let's solve the given equation step-by-step to find out which equation is equivalent to [tex]\(16^{2p} = 32^{p+3}\)[/tex].
1. Rewrite the bases in terms of powers of 2:
[tex]\[
16 = 2^4 \quad \text{and} \quad 32 = 2^5
\][/tex]
2. Substitute these expressions into the equation:
[tex]\[
(2^4)^{2p} = (2^5)^{p+3}
\][/tex]
3. Simplify the exponents using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] for both sides:
[tex]\[
2^{4 \cdot 2p} = 2^{5 \cdot (p+3)}
\][/tex]
[tex]\[
2^{8p} = 2^{5p + 15}
\][/tex]
4. We now have the equation in the form [tex]\(2^{8p} = 2^{5p + 15}\)[/tex]. Since the bases are the same, we can equate the exponents:
[tex]\[
8p = 5p + 15
\][/tex]
5. Solving for [tex]\(p\)[/tex]:
[tex]\[
8p - 5p = 15
\][/tex]
[tex]\[
3p = 15
\][/tex]
[tex]\[
p = 5
\][/tex]
Thus, the equivalent equation to [tex]\(16^{2p} = 32^{p+3}\)[/tex] is [tex]\(2^{8p} = 2^{5p + 15}\)[/tex].
Now let's match this with the given options:
1. [tex]\(8^{4p} = 8^{4p+3}\)[/tex]
2. [tex]\(8^{4p} = 8^{4p+12}\)[/tex]
3. [tex]\(2^{8 \rho} = 2^{5 \rho + 15}\)[/tex]
4. [tex]\(2^{8p} = 2^{5p+3}\)[/tex]
The correct matching option from the list is:
[tex]\[
\boxed{4}
\][/tex]
