Question

If x= 3 + √8 find x4 + 1/ x4​

Answer 1

1154

Step-by-step explanation:

$\blue{\text{\pink{Ni}\red{sa} you can solve this question easily.}}$

If you follow some rules.

Given:

x = 3 + √8

To Find:

${x}^{4} + \dfrac{1}{ {x}^{4} }$

How to Solve:

1. Find the value of 1/x.

2. Than Find x + 1/x

3. Use Algebra Formula

4. Simplify and get answers.

Solution:

$x = 3 + \sqrt{8} \\ \\ \frac{1}{x} = \frac{1}{3 + \sqrt{8} } \\ \\ = \frac{1}{3 + \sqrt{8} } \times \frac{3 - \sqrt{8} }{3 - \sqrt{8} } \: \: [\because(a+b)(a-b) = {a}^{2} - {b}^{2}] \\ \\ = \frac{3 - \sqrt{8} }{ {(3)}^{2} - {( \sqrt{8})}^{2}} \\ \\ = \frac{3 - \sqrt{8} }{9 - 8} \\ \\ = \frac{3 - \sqrt{8} }{1} \\ \\ \therefore \: \: \frac{1}{x} = 3 - \sqrt{8} \\ \\ \tt{Again}, \:\:\: x + \frac{1}{x} = (3 + \sqrt{8}) + (3 - \sqrt{8} ) \\ \\ \tt x + \frac{1}{x} = 6$

Now,

$= {x}^{4} + \frac{1}{ {x}^{4} } \: \: [\because {a}^{2} + {b}^{2} = (a+b)^2 - 2ab]\\ \\ = {( {x}^{2} + \frac{1}{ {x}^{2} } ) }^{2} -2. {x}^{2} .\frac{1}{ {x}^{2} } \\ \\ = { \{ {(x + \frac{1}{x} )}^{2} - 2.x. \frac{1}{x} \}}^{2} - 2 \\ \\ = { \{ {(6)}^{2} - 2\} }^{2} - 2 \\ \\ = {(36 - 2)}^{2} - 2 \\ \\ = 34 {}^{2} - 2 \\ \\ = 1156 - 2 \\ \\ = 1154$

Therefore, Our required answer is 1154.

Answer 2

$\sf{\underline{\underline{\large{\red{Question:-}}}}}$

If x = 3 + √8, find x⁴ + 1/x⁴

$\sf{\underline{\underline{\large{\red{Answer:-}}}}}$

$\mathtt{\blue{\underline{\bold{Finding\:1/x:-}}}}$

x = 3 + √8

➮$\bf{ \frac{1}{x}\: = \: \frac{1}{3 + \sqrt{8} } }$

➮$\bf{ \frac{1}{x } = \frac{1(3 - \sqrt{8)} }{3 + \sqrt{8}(3 - \sqrt{8)} } }$

➮$\bf{ \frac{1}{x} = \frac{3 - \sqrt{8} }{ {(3)}^{2} - { (\sqrt{8} )}^{2} }}$

➮$\bf{ \pink{\frac{1}{x} } = \frac{3 - \sqrt{8} }{9 - 8} =\pink{ 3 - \sqrt{8}} }$

$\mathtt{\blue{\underline{\bold{Finding\:x\:+\:\frac{1}{x}:-}}}}$

➮$\bf{\green{x\: + \:\frac{1}{x}}\: = \:3 \:+\: \sqrt{8}\: +\: 3\: -\: \sqrt{8}\: =\: \green{6}}$

$\mathtt{\blue{\underline{\bold{Finding\:{x}^{2}\:+\:\frac{1}{{x}^{2}}:-}}}}$

➮$\bf{{(\green{x + \frac{1}{x}})}^{2}\: = \:{x}^{2} \:+\: \frac{1}{{x}^{2}}\: + \:(2 * x * \frac{1}{x})}$

➮$\bf{{6}^{2}\: = \:{x}^{2} \:+\: \frac{1}{{x}^{2}}\: + \:2}$

➮$\bf{36\: = \:{x}^{2} \:+\: \frac{1}{{x}^{2}}\: + \:2}$

➮$\bf{\orange{34\: = \:{x}^{2} \:+\: \frac{1}{{x}^{2}}}}$

$\mathtt{\blue{\underline{\bold{Finding\:{x}^{4}\:+\:\frac{1}{{x}^{4}}:-}}}}$

➮$\bf{(\orange{{{x}^{2}+\frac{1}{{x}^{2}}}})^{2}\:=\:{x}^{4}\:+\:\frac{1}{{x}^4}\:+\:(2\:*\:{x}^{2}\:*\:\frac{1}{{x}^2})}$

➮$\bf{{34}^{2}\: = \:{x}^{4} \:+\: \frac{1}{{x}^{4}}\: + \:2}$

➮$\bf{1156\:-2\:={x}^{4}\:+\:\frac{1}{{x}^4}}$

➮$\bf{1154\:={x}^{4}\:+\:\frac{1}{{x}^4}}$

${\underline{\boxed{\green{\bf{Ans.\:=\:1154}}}}}$