Question

If x=233 + 88√7 then what will be the value of x power 1/4 - 3x power -1/4
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Answer 1

$\textbf{Given:}$

$\mathsf{x=233+88\sqrt{7}}$

$\textbf{To find:}$

$\textsf{The value of}\;\mathsf{x^\frac{1}{4}-3\,x^\frac{-1}{4}}$

$\textbf{Solution:}$

$\mathsf{Consider,}$

$\mathsf{x=233+88\sqrt{7}}$

$\mathsf{x=121+112+88\sqrt{7}}$

$\mathsf{x=11^2+(4\sqrt{7})+2(11)(4\sqrt{7})}$

$\mathsf{x=(11+4\sqrt{7})^2}$

$\mathsf{x=(4+7+4\sqrt{7})^2}$

$\mathsf{x=(2^2+\sqrt{7}^2+2(2)(\sqrt{7}))^2}$

$\mathsf{x=((2+\sqrt{7})^2)^2}$

$\mathsf{x=(2+\sqrt{7})^4}$

$\implies\mathsf{x^\frac{1}{4}=2+\sqrt{7}}$

$\mathsf{\dfrac{1}{x^\frac{1}{4}}=\dfrac{1}{2+\sqrt{7}}{\times}\dfrac{2-\sqrt{7}}{2-\sqrt{7}}}$

$\mathsf{\dfrac{1}{x^\frac{1}{4}}=\dfrac{2-\sqrt{7}}{2^2-\sqrt{7}^2}}$

$\mathsf{\dfrac{1}{x^\frac{1}{4}}=\dfrac{2-\sqrt{7}}{4-7}}$

$\mathsf{\dfrac{1}{x^\frac{1}{4}}=\dfrac{2-\sqrt{7}}{-3}}$

$\mathsf{\dfrac{1}{x^\frac{1}{4}}=\dfrac{\sqrt{7}-2}{3}}$

$\mathsf{Now,}$

$\mathsf{x^\frac{1}{4}-3\,x^\frac{-1}{4}}$

$\mathsf{=x^\frac{1}{4}-3\left(\dfrac{1}{x^\frac{1}{4}}\right)}$

$\mathsf{=2+\sqrt{7}-3\left(\dfrac{\sqrt{7}-2}{3}\right)}$

$\mathsf{=2+\sqrt{7}-(\sqrt{7}-2)}$

$\mathsf{=2+\sqrt{7}-\sqrt{7}+2}$

$\mathsf{=4}$

$\implies\boxed{\mathsf{x^\frac{1}{4}-3\,x^\frac{-1}{4}=4}}$

$\textbf{Find more:}$

If x=√5+1÷√5-1 and y=√5-1÷√5+1 find the value of x^2+ xy +y^2

https://brainly.in/question/3728106

if x =root 5 minus root 2 upon root 5 + root 2 and Y is equal to root 5 + root 2 upon root 5 minus root 2 find the value of x square + xy + Y square

https://brainly.in/question/3901417

Answer 2

Step-by-step explanation:

x = 233 + 88sqrt(7)

To find:

The value of x ^ (1/4) - 3x ^ (- 1/4)

Solution:

Consider,

x = 233 + 88sqrt(7)

x = 121 + 112 + 88sqrt(7)

x = 11 ^ 2 + (4sqrt(7)) + 2(11)(4sqrt(7))

x = (11 + 4sqrt(7)) ^ 2

x = (4 + 7 + 4sqrt(7)) ^ 2

x = (2 ^ 2 + sqrt^2 7 + 2(2)(sqrt(7))) ^ 2

x = ((2 + sqrt(7)) ^ 2) ^ 2

x = (2 + sqrt(7)) ^ 4

x₁ = 2 + √7