Question

x-1/x=5 then find x4+1/x4
$x - 1\x = 5 find \: x {}^{4} + 1 \div x {4}^{2}$
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Answer 1

Given:-

• $x - \dfrac{1}{x} = 5$

To find :-

• ${x}^{4} + \dfrac{1}{x ^4}$

Answer:-

Given that,

$x - \dfrac{1}{x} = 5$

Squaring both sides,

$\longrightarrow { \bigg(x - \dfrac{1}{x} \bigg)}^{2} = {5}^{2}$

Expanding L.H.S. using (a - b)² = a² + b² - 2ab, with a = x and b = 1/x,

$\longrightarrow {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \bigg(x \times \dfrac{1}{x} \bigg) = 25$

$\longrightarrow {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 = 25$

$\longrightarrow {x}^{2} + \dfrac{1}{ {x}^{2} } = 27$

Again squaring both sides,

$\longrightarrow { \bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg)}^{2} = {27}^{2}$

Expanding L.H.S. using (a + b)² = a² + b² + 2ab, with a = x² and b = 1/x²,

${\longrightarrow { ({x}^{2}) }^{2} + \dfrac{1}{ ({ {x}^{2}) }^{2} } + 2 \bigg( {x}^{2} \times \dfrac{1}{ {x}^{2} } \bigg) = 729}$

$\longrightarrow {x}^{4} + \dfrac{1}{x ^4} + 2 = 729$

$\longrightarrow \underline{ \underline{{x}^{4} + \dfrac{1}{x ^4} = 727}}$