Which equation is equivalent to [tex]\( 16^{2p} = 32^{p+3} \)[/tex]?

A. [tex]\( 8^{4p} = 8^{4p+3} \)[/tex]
B. [tex]\( 8^{4p} = 8^{4p+12} \)[/tex]
C. [tex]\( 2^{8p} = 2^{5p+15} \)[/tex]
D. [tex]\( 2^{8\rho} = 2^{5\rho+3} \)[/tex]



Answer :

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]