Question

Find x4 + 1/x4 when x+ 1/x = 7

Answer 1

Answer:

${x}^{4} \: + \: \dfrac{1}{ {x}^{4} } \: = \: 2207$

Step-by-step explanation:

Given---->

$x \: + \: \dfrac{1}{x} \: = \: 7$

To find ---->

$value \: of \:$

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

Concept used ---->

$( \: a \: + \: b \: ) ^{2} \: = {a}^{2} \: + \: {b}^{2} \: + \: 2 \: a \: b$

Solution---->

$now$

$x \: + \: \dfrac{1}{x} \: = \: 7$

$squaring \: both \: sides$

$= > \: ( \: x \: + \: \dfrac{1}{x} \: ) ^{2} \: = \: {7}^{2}$

$= > \: {x}^{2} \: + \: \dfrac{1}{ {x}^{2} } \: + \: 2 \: x \: \dfrac{1}{x} \: = \: 49$

$= > \: {x}^{2} \: + \: \dfrac{1}{ {x}^{2} } \: + \: 2 \: = \: 49$

$= > {x}^{2} \: + \: \dfrac{1}{ {x}^{2} } \: = \: 49 \: - \: 2$

$= > \: {x}^{2} \: + \: \dfrac{1}{ {x}^{2} } \: = \: 47$

$squaring \: both \: sides \: again \: we \: get$

$( \: \ {x}^{2} \: + \: \dfrac{1}{ {x}^{2} } \: ) ^{2} = \: ( \: 47 \: ) ^{2}$

$= > {x}^{4} \: + \: \dfrac{1}{ {x}^{4} } \: + \: 2 \: {x}^{2} \: \dfrac{1}{ {x}^{2} } \: = \: 2209$

$= > {x}^{4} \: + \: \dfrac{1}{ {x}^{4} } \: + \: 2 \: = \: 2209$

$= > {x}^{4 \:} \: + \: \dfrac{1}{ {x}^{4} } \: = \: 2209 \: - \: 2$

$= > {x}^{4} \: + \: \dfrac{1}{ {x}^{4} } = \: 2207$