Question

If x-1/x=3, then x⁴+1/x⁴
thank you​

Answer 1

Question :-

$\to \sf{if \: x \: - \frac{1}{x} = 3 \: \: then \: find \: \: {x}^{4} + \frac{1}{ {x}^{4} } }$

Answer :-

$\leadsto \boxed{\sf{ {x}^{4} + \frac{1}{ {x}^{4} } = 119 }} \:$

To Find :-

$\implies \sf{ {x}^{4} + \frac{1}{ {x}^{4} } } \ \:$

Solution :-

We have

$\leadsto \: \sf{x - \frac{1}{x} = 3} \\ \\ \bf{ squaring \: on \: both \: sides \: } \\ \\ \leadsto \sf{} \: { \bigg(x - \frac{1}{x} \bigg)}^{2} = 9 \\ \\ \leadsto \sf{ {x}^{2} + \frac{1}{ {x}^{2} } - 2 \times x \times \frac{1}{ x} = 9} \\ \\ \leadsto \sf{{x}^{2} + \frac{1}{ {x}^{2} } - 2 = 9} \\ \\ \leadsto \sf{ {x}^{2} + \frac{1}{ {x}^{2} } = 9 + 2} \\ \\ \leadsto \sf{ {x}^{2} + \frac{1}{ {x}^{2} } = 11}$

$\bf{again \: squaring \: on \: both \: sides} \\ \\ \leadsto \sf{ { \bigg \{ {x}^{2} + \frac{1}{ {x}^{2} } \bigg \} }^{2} } = 121 \\ \\ \leadsto \sf{ {x}^{4} + \frac{1}{ {x}^{4} } + 2 \times {x}^{2} \times \frac{1}{ {x}^{2} } = 121 } \\ \\ \leadsto \sf{ {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 121} \\ \\ \leadsto \sf{ {x}^{4} + \frac{1}{ {x}^{4} } = 121 - 2} \\ \\ \leadsto \boxed{\sf{ {x}^{4} + \frac{1}{ {x}^{4} } = 119 }}$

Answer 2

Step-by-step explanation:

$\bold{Given - } \\ x - \frac{1}{x} = 3 \\ \\ \bold{To \: find - } \\ value \: of \: {x}^{4} + \frac{1}{ {x}^{4} } \\ \\ \bold{Now,} \\According \: to \: the \: question \\ \bold{squaring \: both \: sides - } \\ \implies (x - \frac{1}{x}) {}^{2} = (3) {}^{2} \\ \implies{(x)}^{2} + {( \frac{1}{x}) }^{2} - 2 \times \cancel{x} \times \frac{1}{ \cancel{x}} = 9 \\ \implies {x}^{2} + \frac{1}{ {x}^{2} } = 9 + 2 \\ \implies {x}^{2} + \frac{1}{ {x}^{2} } = 11 \\ \bold{Again \: squaring \: both \: sides - } \\ \implies( {x}^{2} + \frac{1}{ {x}^{2} } ) {}^{2} = (11) {}^{2} \\ \implies {x}^{4} + \frac{1}{ {x}^{4} } + 2 \times \cancel{ {x}^{2} }\times \frac{1}{ \cancel{ {x}^{2}} } = 121 \\ \implies {x}^{4} + \frac{1}{ {x}^{4} } = 121 - 2 \\ \implies {x}^{4} + \frac{1}{ {x}^{4} } = 119 \\ \\ Hence, \\ \bold{The \: value \: of \: {x}^{4} + \frac{1}{ {x}^{4} } \: is \: 119} \\ \\ \bold{Formula \: used - } \\ (a + b) {}^{2} = {a}^{2} + 2ab + {b}^{2} \\ (a - b) {}^{2} = {a}^{2} - 2ab + {b}^{2}$

Other formulas like this -

• (a+b)(a-b) = a² - b²
• (a+b)³ = a³ + 3a²b + 3b²a + b³
• (a-b)³ = a³ - 3a²b + 3b²a - b³