To determine which equation is equivalent to [tex]\(16^{2p} = 32^{p+3}\)[/tex], let's start by rewriting the given equation using the same base.
We know that:
- [tex]\(16 = 2^4\)[/tex]
- [tex]\(32 = 2^5\)[/tex]
Rewrite [tex]\(16^{2p}\)[/tex] and [tex]\(32^{p+3}\)[/tex] using base 2:
[tex]\[
16^{2p} = (2^4)^{2p}
\][/tex]
[tex]\[
32^{p+3} = (2^5)^{p+3}
\][/tex]
Now simplify each side:
[tex]\[
(2^4)^{2p} = 2^{4 \cdot 2p} = 2^{8p}
\][/tex]
[tex]\[
(2^5)^{p+3} = 2^{5 \cdot (p + 3)} = 2^{5p + 15}
\][/tex]
So the original equation [tex]\(16^{2p} = 32^{p+3}\)[/tex] simplifies to:
[tex]\[
2^{8p} = 2^{5p + 15}
\][/tex]
Now look at the given options to see which matches this result:
1. [tex]\(8^{4p} = 8^{4p+3}\)[/tex]
2. [tex]\(8^{4p} = 8^{4p+12}\)[/tex]
3. [tex]\(2^{8p} = 2^{5p+15}\)[/tex]
4. [tex]\(2^{8\rho} = 2^{5\rho+3}\)[/tex]
The correct equivalent equation is:
[tex]\[
2^{8p} = 2^{5p + 15}
\][/tex]
Thus, the equivalent equation among the options given is:
[tex]\[
2^{8p} = 2^{5p + 15}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{2^{8p} = 2^{5p + 15}}
\][/tex]
