$p$ 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 $p$.

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

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

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

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

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

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 $\dfrac{1}{2}$.

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

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

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

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

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

Latest Math Topics

Latest Math Problems

Email subscription

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.