Respuesta :

[tex]x + \frac{1}{x} = \frac{5}{2} \\ = > {(x + \frac{1}{x} )}^{2} = {( \frac{5}{2} )}^{2} \\ = > {x}^{2} + 2 \times x \times \frac{1}{x} + \frac{1}{ {x}^{2} } = \frac{25}{4} \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = \frac{25}{4} - 2 \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = \frac{25 - 8}{4} \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = \frac{17}{4} [/tex]

[tex] {x}^{2} + \frac{1}{ {x}^{2} } = \frac{17}{4} \\ = > {( {x}^{2} + \frac{1}{ {x}^{2} } )}^{2} = {( \frac{17}{4} )}^{2} \\ = > {x}^{4} + 2 \times {x}^{2} \times \frac{1}{ {x}^{2} } + \frac{1}{ {x}^{4} } = \frac{289}{16} \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = \frac{289}{16} - 2 \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = \frac{289 - 32}{16} \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = \frac{257}{16} [/tex]

Hope you could understand.

If you have any query, feel free to ask.

SalmonWorldTopper

Step-by-step explanation:

We have: {x+(1/x)} = 5/2

On, squaring on both sides,we get

⇛{x+(1/x)}² = (5/2)²

Comparing the given expression with (a+b)², we get

a = x and b = 1/x

Now, using (a+b)² = a²+b²+2ab, we get

⇛x² + (1/x)² + 2(x)(1/x) = (5/2)²

Both x will cancel out because they are in multiple sign.

⇛x² + (1/x²) + 2 = {(5*5)/(2*2)}

⇛x² + (1/x²) + 2 = 25/4

Shift the number 2 from LHS to RHS, changing it's sign.

⇛x² + (1/x²) = (25/4) - 2

⇛x² + (1/x²) = (25/4 ) - (2/1)

Take the LCM of the denominator 4 & 1 is 4 is RHS.

⇛x² + (1/x²) = {(25*1 - 2*4)/4)

⇛x² + (1/x²) = {(25-8)/4}

⇛x² + (1/x²) = (17/4)

Again, squaring on both sides, we get

{x² + (1/x²)}² = (17/4)²

Comparing the given expression with (a+b)², we get

a = x² and b = (1/x²)

Now, using (a+b)² = a²+b²+2ab, we get

⇛(x²)² + (1/x²)² + 2(x²)(1/x²) = (17/4)²

⇛x⁴ + (1/x⁴) + 2(x²)(1/x²) = (17/4)²

Both x² will cancel out because they are in multiple sign.

⇛x⁴ + (1/x⁴) + 2 = {(17*17)/(4*4)}

⇛x⁴ + (1/x⁴) + 2 = (289/16)

Shift the number 2 from LHS to RHS, changing it's sign.

⇛x⁴ + (1/x⁴) = (289/16) - 2

⇛x⁴ + (1/x⁴) = (289/16) - (2/1)

Take the LCM of 16 and 1 is 16 in RHS.

⇛x⁴ + (1/x⁴) = {(289*1 - 2*16)/16}

⇛x⁴ + (1/x⁴) = {(289-32)/16}

⇛x⁴ + (1/x⁴) = (257/16)

Therefore, x⁴ + (1/x⁴) = 257/16

Answer: Hence, the value of x⁴ + (1/x⁴) is 257/16.

Please let me know if you have any other questions or doubt in my explanation.

Otras preguntas