Question

Determine the value of ${{\left( 8x \right)}^{x}}$ , If ${{9}^{x+2}}=240+{{9}^{x}}$

Answer 1

Hint: Here we have been given an equation in exponent form and we have to find the value of the term given. Firstly we will use the property of exponent and separate the terms on the left hand side. Then we will bring the same terms on one side and take common. Finally we will simplify the equation obtained and get the desired answer.

Complete step-by-step solution:
The equation is given as follows:
${{9}^{x+2}}=240+{{9}^{x}}$…..$\left( 1 \right)$
We have to find the value of,
${{\left( 8x \right)}^{x}}$ ……$\left( 2 \right)$
Now as we know that,
${{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}$
Using above formula in equation (1) we get,
${{9}^{x}}\times {{9}^{2}}=240+{{9}^{x}}$
$\Rightarrow {{9}^{x}}\times 81=240+{{9}^{x}}$
Taking the exponent values on one side we get,
$\Rightarrow {{9}^{x}}\times 81-{{9}^{x}}=240$
$\Rightarrow {{9}^{x}}\left( 81-1 \right)=240$
Simplifying further we get,
$\Rightarrow {{9}^{x}}\times 80=240$
$\Rightarrow {{9}^{x}}=\dfrac{240}{80}$
Dividing the numerator by the denominator on the right side we get,
$\Rightarrow {{9}^{x}}=3$….$\left( 3 \right)$
Now as know that,
${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$
Using in equation (3) we get,
$\Rightarrow {{\left( {{3}^{2}} \right)}^{x}}=3$
$\Rightarrow {{3}^{2x}}={{3}^{1}}$
As the base of the exponent is same on comparing we get,
$2x=1$
$\Rightarrow x=\dfrac{1}{2}$
Put the above value in equation (2) we get,
${{\left( 8x \right)}^{x}}={{\left( 8\times \dfrac{1}{2} \right)}^{\dfrac{1}{2}}}$
$\Rightarrow {{\left( 8x \right)}^{x}}={{\left( 4 \right)}^{\dfrac{1}{2}}}$
As we know the square root of $4$ is $2$ we get,
${{\left( 8x \right)}^{x}}=2$
Hence ${{\left( 8x \right)}^{x}}$ is equal to $2$.

Note: Exponent simply means multiplying the term by itself that many times. It is an easier way to write down a value when the same number is multiplied by itself many times. Exponent is expressed as ${{a}^{b}}$ where $a$ is the base and $b$ is the power. There are laws of exponent firstly if the same base exponent is multiplied there power gets added up if the same base exponent is divided the power gets subtracted and if the power of any exponent is in negative then we just find the reciprocal of that value.