Question

x+1/x=4 the find the value of x^4+ 1/x^4​

Answer 1

EXPLANATION.

→ x + 1/x = 4 ..... (1)

→ To find the value of x⁴ + 1/x⁴.

→ squaring on both sides of equation (1)

we get,

→ ( x + 1/x )² = (4)²

→ x² + 1/x² + 2 X ( x) X ( 1/x ) = 16

→ x² + 1/x² + 2 = 16

→ x² + 1/x² = 14 .......(2)

→ From equation (2) again squaring

we get,

→ ( x² + 1/x² )² = ( 14 )²

→ ( x⁴ + 1/x⁴ + 2 X (x²) X (1/x²) = 196

→ ( x⁴ + 1/x⁴ + 2 ) = 196

→ x⁴ + 1/x⁴ = 196 - 4

→ x⁴ + 1/x⁴ = 194

Answer 2

$x + \frac{1}{x} = 4$

$\huge {\sf \: \blue{square \: both \: sides}}$

$(x + \frac{1}{n} {)}^{2} = {4}^{2}$

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

$\green{a = x \: \: \: \: \: \: \: \: \: \: \: \: \: \: b = \frac{1}{n} }$

${x}^{2} + 2x \times x \times \frac{1}{x} + \frac{1}{x} \: 2 = 16$

${x}^{2} + \frac{1}{ {x}^{2} } + 2 = 16$

${x}^{2} + \frac{1}{ {x}^{2} } = 14$

$\huge{\sf \red{square \: both \: side}}$

$⇝ \: \: ( {x}^{2} + \frac{1}{ {n}^{2} } {)}^{2} = {14}^{2}$

$⇝ \: \: {x}^{4} + 2 \times { \cancel{x}}^{2} \times \frac{1}{ { \cancel{x}}^{2} } + \frac{1}{ {x}^{4} } = 196$

$⇝ \: \: {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 196$

$⇝ \: \: {x}^{4} + \frac{1}{ {x}^{4} } = 196 - 2 = 194$
$\red{ \boxed{ \pink{⇝194}}}$