Question

Consider the following expression and find the value of ‘x’: $8{{x}^{\dfrac{3}{2n}}}-8{{x}^{-\dfrac{3}{2n}}}=63$
This question has multiple correct options.
A. ${{2}^{2n}}$
B. ${{2}^{\dfrac{1}{2n}}}$
C. ${{2}^{3n}}$
D. ${{2}^{\dfrac{1}{3n}}}$

Answer 1

First consider the ${{x}^{\dfrac{3}{2n}}}=y$by replacing this value by y then we will get quadratic equation in y. solve the quadratic equation in y then we will get values of y and again replace y by corresponding value in x then we will get the value of x.

Complete step by step answer:
Given that $8{{x}^{\dfrac{3}{2n}}}-8{{x}^{-\dfrac{3}{2n}}}=63$
For this problem we have to find the value of x
Let us consider ${{x}^{\dfrac{3}{2n}}}=y$
Replacing that value by x then we will get the equation as follows
$8y-8{{y}^{-1}}=63$
$8y-\dfrac{8}{y}=63$
$8{{y}^{2}}-8=63y$
$8{{y}^{2}}-63y-8=0$. . . . . . . . . . . (1)
Now solve the above equation quadratic equation then we will get values of y as follows
$8{{y}^{2}}-64y+y-8=0$
$8y\left( y-8 \right)+1\left( y-8 \right)=0$
$\left( y-8 \right)\left( 8y+1 \right)=0$
$y=8,y=-\dfrac{1}{8}$
So the obtained values of y is $y=8,y=-\dfrac{1}{8}$. . . . . . . . . . (2)
First we have considered that ${{x}^{\dfrac{3}{2n}}}=y$
Let us take $y=8$
${{x}^{\dfrac{3}{2n}}}=8$
8 can be written 2 to the power of 3
${{x}^{\dfrac{3}{2n}}}={{2}^{3}}$
$x={{({{2}^{3}})}^{\dfrac{2n}{3}}}$
$x={{2}^{2n}}$. . . . . . . . . . . (3)
So the obtained value of x for y=8 is $x={{2}^{2n}}$
Let us take $y=-\dfrac{1}{8}$
$\dfrac{1}{8}$can be 2 to the power of -3
${{x}^{\dfrac{3}{2n}}}=-\dfrac{1}{8}$
${{x}^{\dfrac{3}{2n}}}=-{{2}^{-3}}$
$x={{(-{{2}^{-3}})}^{\dfrac{2n}{3}}}$
$x={{\left( -2 \right)}^{-2n}}$
$x={{\left( 2 \right)}^{-2n}}$
$x={{\left( 2 \right)}^{\dfrac{1}{2n}}}$. . . . . . . . . . . (4)
So the obtained value of x for $y=-\dfrac{1}{8}$is $x={{\left( 2 \right)}^{\dfrac{1}{2n}}}$
So the correct option is option (A), (B).

Note:
In this problem we will get quadratic in y so we can get two values of y , we have considered a corresponding value in x by y so we also get two of values of x. we used factorization method to solve quadratic equation in y we can get the two values of y either by factorization method or using formula.