If $9^{p+2} -9^p = 240$, find the value of ${(8p)}^p$

$p$ is a literal and it involved in forming an expression in terms of number $9$ to represent the number $240$. The solution of the equation that contains exponential terms gives the value of $p$ and it is useful to find the value of ${(8p)}^p$ in this maths problem.

Firstly, let us solve this equation to obtain the value of $p$.

$9^{p+2} -9^p = 240$

Step: 1

Use the product rule of exponents to express the exponential term $9^{p+2}$ as the product of two exponential terms having same base.

$\implies 9^p \times 9^2 -9^p = 240$

Step: 2

Write the value of $9$ squared in number form.

$\implies 9^p \times 81 -9^p = 240$

Step: 3

Simplify the left hand side of the equation to obtain the value of $p$.

$\implies 81 \times 9^p -9^p = 240$

$\implies 9^p(81-1) = 240$

$\implies 9^p \times 80 = 240$

$\implies 9^p = \dfrac{240}{80}$

$\implies 9^p = 3$

$\implies {(3^2)}^p = 3$

Apply the power law of exponent of an exponential term.

$\implies 3^{2p} = 3$

$\implies 3^{2p} = 3^1$

The bases of both sides of this equation is same. So, their exponents should also be equal mathematically.

$\implies 2p = 1$

$\therefore \,\,\,\,\,\, p = \dfrac{1}{2}$

Therefore, the given equation is solved and the value of $p$ is $\dfrac{1}{2}$.

Step: 4

Substitute the value of $p$ in ${(8p)}^p$ to obtain its value.

${(8p)}^p = {(8 \times \dfrac{1}{2})}^{\frac{1}{2}}$

$\implies {(8p)}^p = \sqrt{\dfrac{8}{2}}$

$\implies {(8p)}^p = \sqrt{4}$

$\therefore \,\,\,\,\,\, {(8p)}^p = 2$