Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
solving exponential equations by rewriting the base
which equation is equivalent to ( 16^{2p} = 32^{p + 3} )?
( 2^{8p} = 2^{5p + 3} )
( 8^{4p} = 8^{4p + 3} )
( 8^{4p} = 8^{4p + 12} )
( 2^{8p} = 2^{5p + 15} )

Explanation:

Step1: Rewrite 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 the property \((a^m)^n=a^{mn}\).
For the left - hand side: \((2^4)^{2p}=2^{4\times2p}=2^{8p}\)
For the right - hand side: \((2^5)^{p + 3}=2^{5\times(p + 3)}=2^{5p+15}\)
So the equation \(16^{2p}=32^{p + 3}\) is equivalent to \(2^{8p}=2^{5p + 15}\)

Let's check the other options by rewriting their bases:

• Option with \(8^{4p}=8^{4p + 3}\): We know that \(8 = 2^3\), so \(8^{4p}=(2^3)^{4p}=2^{12p}\) and \(8^{4p + 3}=(2^3)^{4p+3}=2^{12p + 9}\), which is not equivalent to \(2^{8p}=2^{5p + 15}\)
• Option with \(8^{4p}=8^{4p + 12}\): \(8^{4p}=(2^3)^{4p}=2^{12p}\) and \(8^{4p+12}=(2^3)^{4p + 12}=2^{12p+36}\), not equivalent.
• Option with \(2^{8p}=2^{5p + 3}\): The exponent on the right - hand side is \(5p + 3\) instead of \(5p+15\), so it's not equivalent.

Answer:

\(2^{8p}=2^{5p + 15}\) (the last option in the list of given options)