Question

x+1/x=3 then x^5+1/x^5=?

Answer 1

Heya !!

Here's your answer.. ⬇⬇
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$x + \frac{1}{x} = 3 \\ \\ {(x + \frac{1}{x} )}^{2} = {(3)}^{2} \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 9 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 9 - 2 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 7 \\ \\ {(x + \frac{1}{x} })^{3} = {(3)}^{3} \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3x \times \frac{1}{x} \times (x + \frac{1}{x} ) = 27\\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3(3) = 27\\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = 27 - 9 \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = 18 \\ \\ {( {x}^{2} + \frac{1}{ {x}^{2} }) }^{2} = {(7)}^{2} \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 49 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 49 - 2 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 47 \\ \\ ( {x}^{4} + \frac{1}{ {x}^{4} } ) \times (x + \frac{1}{x} ) = 47 \times 3 \\ \\ {x}^{5} + \frac{x}{ {x}^{4} } + \frac{ {x}^{4} }{x} + \frac{1}{ {x}^{5} } = 141 \\ \\ {x}^{5} + \frac{1}{ {x}^{5} } + {x}^{3} + \frac{1}{ {x}^{3} } = 141 \\ \\ {x}^{5} + \frac{1}{ {x}^{5} } + 18 = 141 \\ \\ {x}^{5} + \frac{1}{ {x}^{5} } = 141 - 18 \\ \\ {x}^{5} + \frac{1}{ {x}^{5} } = 123$
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Hope it helps..

Thanks :)

Answer 2

HELLO DEAR,

given that:-

(x + 1/x) = 3

now,

on squaring both side,

(x + 1/x)² = 3²

⇒(x² + 1/x² + 2 * x * 1/x) = 9
∴ [ (a + b)² = (a² + b² + 2ab) ]

⇒(x² + 1/x²) = 9 - 2

now,

⇒(x + 1/x) = 3
∴ [ on cubing both side ]

⇒(x + 1/x)³ = 3³

⇒[ x³ + 1/x³ + 3*x*1/x(x + 1/x) = 27
∴ [ (a + b)³ = a³ + b³ + 3ab(a + b) ]

⇒x³ + 1/x³ + 3(3) = 27
∴ [ (x + 1/x) = 3 ]

⇒(x³ + 1/x³) = 27 - 9

⇒(x³ + 1/x³) = 18

∴ [ multiply Both side by (x + 1/x) ]

⇒(x³ + 1/x³)(x² + 1/x²) = 18(x² + 1/x²)

⇒(x³x² + x³/x² + x²/x³ + 1/x³ * 1/x²) = 18 * 7

⇒(x⁵ + x + 1/x + 1/x⁵) = 126

⇒(x + 1/x) + (x⁵ + 1/x⁵) = 126
∴ [ (x + 1/x) = 3]

⇒(x⁵ + 1/x⁵) = 126 - 3

⇒(x⁵ + 1/x⁵) = 123

I HOPE ITS HELP YOU DEAR,
THANKS