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The equations below relate the distinct positive reabrumbers $a$ , $b$ , $c$ , and $d$ . Which equation correctly expresses $b$ in terms of $d$ ? $\newline$ $\begin{cases} c = 2a^{2} \ a^{3b} = \left(\frac{c}{2}\right)^{d+1} \end{cases}$

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Compare linear and exponential growth

Full solution

Q. The equations below relate the distinct positive reabrumbers

a

,

b

,

c

, and

d

. Which equation correctly expresses

b

in terms of

d

?

\newline

\begin{cases} c = 2a^{2} \ a^{3b} = \left(\frac{c}{2}\right)^{d+1} \end{cases}

Given equations: Given equations: $\newline$ $c = 2a^2$ $\newline$ $a^{3b} = \left(\frac{c}{2}\right)^{d+1}$
Substitute and simplify: $\newline$ Step $2$ : Substitute $c = 2a^2$ into the second equation: $\newline$ $a^{3b} = \left(\frac{2a^2}{2}\right)^{d+1}$ $\newline$ $a^{3b} = (a^2)^{d+1}$
Simplify right side: $\newline$ Step $3$ : Simplify the right side: $\newline$ $a^{3b} = a^{2(d+1)}$
Set exponents equal: $\newline$ Step $4$ : Since the bases are the same, set the exponents equal: $\newline$ $3b = 2(d+1)$
Solve for b: $\newline$ Step $5$ : Solve for $b$ : $\newline$ $3b = 2d + 2$ $\newline$ $b = \frac{2d + 2}{3}$

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