Question

which equation is equivalent to $16^{2p} = 32^{p + 3}$?
\bigcirc\\ $8^{4p} = 8^{4p + 3}$
\bigcirc\\ $8^{4p} = 8^{4p + 12}$
\bigcirc\\ $2^{8p} = 2^{5p + 15}$
\bigcirc\\ $2^{8p} = 2^{5p + 3}$

Explanation:

Step1: Express bases as powers of 2

We know that \(16 = 2^4\) and \(32 = 2^5\). So we can rewrite the given equation \(16^{2p}=32^{p + 3}\) using these.
For the left - hand side: \((2^4)^{2p}\), by the power - of - a - power rule \((a^m)^n=a^{mn}\), we have \((2^4)^{2p}=2^{4\times2p}=2^{8p}\).
For the right - hand side: \((2^5)^{p + 3}\), using the power - of - a - power rule \((a^m)^n=a^{mn}\), we get \((2^5)^{p + 3}=2^{5\times(p + 3)}=2^{5p+15}\).

So the equation \(16^{2p}=32^{p + 3}\) is equivalent to \(2^{8p}=2^{5p + 15}\).

Step2: Analyze other options (optional but for verification)

• Option 1: \(8 = 2^3\), \(8^{4p}=(2^3)^{4p}=2^{12p}\), \(8^{4p + 3}=(2^3)^{4p+3}=2^{12p + 9}\), which is not equal to \(2^{8p}\) and \(2^{5p+15}\) in general.
• Option 2: \(8^{4p}=(2^3)^{4p}=2^{12p}\), \(8^{4p + 12}=(2^3)^{4p+12}=2^{12p+36}\), not equal to \(2^{8p}\) and \(2^{5p + 15}\) in general.
• Option 4: \(2^{5p+3}\) is not equal to \(2^{5p + 15}\) (since the exponents \(5p + 3

eq5p+15\) for real \(p\)).

Answer:

\(2^{8p}=2^{5p + 15}\) (the third option: \(2^{8p}=2^{5p+15}\))