Question

If x = 5+ 2√6, then the value of ( x + 1/x) ^4 is​

Answer 1

Step-by-step explanation:

x=5+2√6

1/x=5-2√6/(5+2√6)(5-2√6)

=5-2√6/25-24=5-2√6

x+1/x=5+2√6+5-2√6=10

(x+1/x)^4=(10)^4=10000

Answer 2

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:x = 5 + 2 \sqrt{6}$

Now,

Consider,

$\rm :\longmapsto\:\dfrac{1}{x}$

$\rm \: \: = \: \dfrac{1}{5 + 2 \sqrt{6} }$

On rationalizing the denominator, we get

$\rm \: \: = \: \dfrac{1}{5 + 2 \sqrt{6} } \times \dfrac{5 - 2 \sqrt{6} }{5 - 2 \sqrt{6} }$

$\rm \: \: = \: \dfrac{5 - 2 \sqrt{6} }{ {(5)}^{2} - {(2 \sqrt{6})}^{2} }$

$\: \: \: \: \: \: \: \green{\boxed{ \bf{ \: \because \: (x + y)(x - y) = {x}^{2} - {y}^{2}}}}$

$\rm \: \: = \: \dfrac{5 - 2 \sqrt{6} }{25 - 24}$

$\rm \: \: = \: 5 - 2 \sqrt{6}$

$\bf\implies \:\dfrac{1}{x} = 5 - 2 \sqrt{6}$

Consider,

$\green{\bf :\longmapsto\:x + \dfrac{1}{x}}$

$\rm \: \: = \: 5 + 2 \sqrt{6} + 5 - 2 \sqrt{6}$

$\rm \: \: = \: 10$

$\bf\implies \:x + \dfrac{1}{x} = 10$

Hence,

$\rm :\longmapsto\: {\bigg(x + \dfrac{1}{x} \bigg) }^{4}$

$\rm \: \: = \: {(10)}^{4}$

$\rm \: \: = \: 10000$

$\purple{ \boxed{\bf\implies \:\: {\bigg(x + \dfrac{1}{x} \bigg) }^{4} = 10000}}$

Additional Information :-

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)