Question

If x+1\x=5 find the value of x power 4+1/x power 4

Answer 1

Answer:

527

Step-by-step explanation:

Given---> x + 1 / x = 5

To find ---> Value of ( x⁴ + 1 / x⁴ )

Solution---> We have an identity

( a + b )² = a² + b² + 2a b

Now , putting a= x and b= 1 / x we get

( x + 1 / x )² = x² + 1 / x² + 2 x (1 / x)

=> ( 5 )² = x² + 1 / x² + 2

=> 25 = x² + 1 / x² + 2

=> 25 - 2 = x² + 1 / x²

=> x² + 1 / x² = 23

Squaring both sides we get

=> ( x² + 1 / x² )² = 529

Again applying formula of (a + b )² we get

=> ( x² )² + ( 1 / x² )² + 2 x² ( 1 / x² ) = 529

=> x⁴ + 1 / x⁴ + 2 = 529

=> x⁴ + 1 / x⁴ = 529 - 2

=> x⁴ + 1 / x⁴ = 527

Answer 2

To find value of given expression firstly we have to find the value of x .

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GIVEN THAT,

$\frac{x + 1}{x} = 5 \\ \\ x + 1 = 5x \\ \\ 1 = 4x \\ \\ \boxed{\boxed{{x = \frac{1}{4} }}}$

Now,

using above value of x we can easily calculate the value of given expression as:-

${x}^{4} + ( { \dfrac{1}{x}) }^{4} \\ \\ = ( {\frac{1}{4} )}^{4} + {4}^{4} \\ \\ = \frac{1}{256} + 256 \\ \\ = \frac{1 + 65536}{256} \\ \\ = \frac{65537}{256}$

So, the value of given expression is 65537/256 .