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

Hint: 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.