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Q.

If it is given that ( $x + \frac{1}{x}$ ) =4 find the value of ( $x^{4} + \frac{1}{x^{4}}$ )

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a

120

b

194

c

160

d

140

answer is B.

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Detailed Solution

Concept- Calculating the total of both terms raised to the power of four is necessary. In order to do that, we must first square both sides of the equation and solve it in order to obtain the sum (

x^{2} + \frac{1}{x^{2}}

). Since the terms in (x+

\frac{1}{x}

) are reciprocal of one another, the ′2ab′ term in

{(a + b)}^{2}

will be canceled, leaving only ′2′. Finally, we must square both sides of the equation once more and solve in order to obtain the value of the required sum in a manner similar to that described above.
Given, (x+

\frac{1}{x}

) =4

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

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

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

x^{2} + \frac{1}{x^{2}} = 14_____(1)

Squaring equation 1

({x^{2} + \frac{1}{x^{2}})}^{2} = 14^{2}

{(x^{2})}^{2} + {(\frac{1}{x^{2}})}^{2}

=

14^{2}

{(x^{2})}^{2} + {(\frac{1}{x^{2}})}^{2} + 2 . x^{2} . \frac{1}{x^{2}} = 196

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

x^{4} + \frac{1}{x^{4}} = 194

Hence, the correct answer is option 2) 194.

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