Question

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

Explanation:

Step1: Rewrite bases as powers of 2

$16=2^4$, $32=2^5$, so substitute:
$$(2^4)^{2p} = (2^5)^{p+3}$$

Step2: Apply exponent power rule

Use $(a^m)^n=a^{m \cdot n}$:
$$2^{8p} = 2^{5(p+3)}$$

Step3: Simplify right-hand exponent

Distribute 5 to the terms in parentheses:
$$2^{8p} = 2^{5p+15}$$

Answer:

$\boldsymbol{2^{8p} = 2^{5p+15}}$