Question

If X square + 1 by x square is equal to 2 then find the value of x to the power 4 + 1 by x to the power 4

Answer 1

$\sf\large\underline{Given:}$

• $\rm{x^2+\dfrac{1}{x^2}=2}$

$\sf\large\underline{To\: Find:}$

• $\rm{x^4+\dfrac{1}{x^4}=?}$

$\sf\large\underline{Solution:}$

• Simply applying identity here to calculate value]

$\sf\large\underline{Identity\:Used:}$

$\tt{\implies (a+b)^2=a^2+b^2+2ab}$

$\tt{\implies a^2+b^2=(a+b)^2-2ab}$

$\rm{\implies x^2+\dfrac{1}{x^2}=2}$

• Squaring on both sides here]

$\tt{\implies (x^2+\dfrac{1}{x^2})^2=2^2}$

$\tt{\implies x^4+\dfrac{1}{x^4}+2=4}$

$\tt{\implies x^4+\dfrac{1}{x^4}=4-2}$

$\tt{\implies x^4+\dfrac{1}{x^4}=2}$

$\sf\large{Hence,}$

$\rm{\implies x^4+\dfrac{1}{x^4}=2}$

$\sf\large\underline{Some\:important\: Formula:}$

$\tt{\implies (a+b)^3=a^3+3a^2b+3ab^2+b^3}$

$\tt{\implies a^3+b^3=(a+b)^3-3ab(a+b)}$

$\tt{\implies a^2-b^2=(a+b)(a-b)}$

Answer 2

○ Question ○

If X square + 1 by x square is equal to 2 then find the value of x to the power 4 + 1 by x to the power 4 .

$\rule{300}2$

Given :- $x^2+\frac{1}{x^2}=2\\ \implies x^4+\frac{1}{x^4}=?$

◇ Answer ◇

$\rightarrow x^2+\frac{1}{x^2}=2 [\: squaring\:on\:both\:side]\\ \rightarrow (x^2+\frac{1}{x^2}^2)=2^2\\ \rightarrow x^4+\frac{1}{x^4}=4\\ \rightarrow x^4+\frac{1}{x^4}-2×x×\frac{1}{x}=4\\ \rightarrow x^4+\frac{1}{x^4}-2=4\\ \rightarrow x^4+\frac{1}{x^4}=4+2 \\ \rightarrow\red{\fbox{ x^4+\frac{1}{x^4}=6}}$

★ Identity used in solution

$\leadsto\pink{\fbox{\fbox { (x+y)^2=(x^2+2xy+y^2)}}}\\ \leadsto\pink{\fbox{\fbox{ (x+y)^2=(x+y)^2-2xy)}}}$