Question
Find the value of $(8p)^p$ if $9^{p + 2} - 9^p = 240.$
✓
Answer
$9 p+2-9 p=240$
$\Rightarrow 9^p\left(9^2-1\right)=240$
$\Rightarrow 9^p(80)=240$
$\Rightarrow 9^p=3$
$\Rightarrow 3^{2 p}=3$
$\Rightarrow 2 p=1$
$\Rightarrow p=\frac{1}{2}$
$(8 p)^p=\left(2^3 p\right)^p$
$=\left(2^3 \cdot \frac{1}{2}\right)^{\frac{1}{2}}$
$=\left(2^{3-1}\right)^{\frac{1}{2}}$
$=\left(2^2\right)^{\frac{1}{2}}$
$=2 .$
$\Rightarrow 9^p\left(9^2-1\right)=240$
$\Rightarrow 9^p(80)=240$
$\Rightarrow 9^p=3$
$\Rightarrow 3^{2 p}=3$
$\Rightarrow 2 p=1$
$\Rightarrow p=\frac{1}{2}$
$(8 p)^p=\left(2^3 p\right)^p$
$=\left(2^3 \cdot \frac{1}{2}\right)^{\frac{1}{2}}$
$=\left(2^{3-1}\right)^{\frac{1}{2}}$
$=\left(2^2\right)^{\frac{1}{2}}$
$=2 .$
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