Question

X-1/X=√5 find the value of x^4-1/x^4

Answer 1

Answer:

25

Step-by-step explanation:

X-(1/X) =√5

now, to find the value of x^4-1/X^4

we will raise the power to 4 on both sides

So,

it will become

x^4-1/x^4 = √ 5^4

so x^4-1/x^4 =25

Answer 2

Answer:

Take square of first equation

$(x - \frac{1}{x} ) {}^{2} = 5 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } - 2(x)( \frac{1}{x} ) = 5 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 5 + 2 = 7$

Now note that

$(x + \frac{1}{x} ) {}^{2} = (x - \frac{1}{x} ) {}^{2} + 4(x)( \frac{1}{x} ) \\ \\( x + \frac{1}{x} ) {}^{2} = 5 + 4 = 9 \\ \\ x + \frac{1}{x} = 3$

To reduce complexity assume x > 0.

Now

${x}^{4} - \frac{1}{ {x}^{4} } \\ \\ = ({x}^{2} ) {}^{2} - ( \frac{1}{ {x}^{2} } ) {2}^{2} \\ \\ = ( {x}^{2} + \frac{1}{ {x}^{2} } )( {x}^{2} - \frac{1}{ {x}^{2} } ) \\ \\ = ( {x}^{2} + \frac{1}{ {x}^{2} } )(x + \frac{1}{x} )(x - \frac{1}{x} ) \\ \\ put \: values \: from \: above \\ = 7 \times 3 \times \sqrt{5} \\ = 21 \sqrt{5}$