Question

if x= 1-root 2 find the value of x-1uponx whole square

Answer 1

Answer:

4

Step-by-step explanation:

As per the provided information in the given question, we have :

$\bull \; \sf { x = 1 - \sqrt{2} }$

We are asked to calculate the value of,

$\bull \; \sf { { \Bigg ( x - \dfrac{1}{x} \Bigg )}^{2} }$

Here, 1/x is the reciprocal of x. So,

$\longmapsto \sf { \dfrac{ 1}{x} = \dfrac{1}{1-\sqrt{2} }}$

Now, we have to rationalize the denominator. In order to rationalize the denominator, we multiply the rationalising factor of the denominator with both the numerator and the denominator of the fraction.Here, denominator is in the form of (a - b).

• Rationalising factor of (a - b) is (a + b)

Thus , rationalising factor of (1 - √2) is (1 + √2). Multiplying (1 + √2) with both the numerator and the denominator of the fraction.

$\longmapsto \sf { \dfrac{ 1}{x} = \dfrac{1}{1-\sqrt{2}} \times \dfrac{1+ \sqrt{2}}{1+\sqrt{2} } }$

Rearranging the terms.

$\longmapsto \sf { \dfrac{ 1}{x} = \dfrac{1(1+ \sqrt{2})}{(1-\sqrt{2})(1+\sqrt{2}) }}$

Using the identity,

• (a + b)(a - b) = a² - b²

$\longmapsto \sf { \dfrac{ 1}{x} = \dfrac{1+ \sqrt{2} }{(1)^2 - (\sqrt{2})^2}}$

Writing the squares in the denominator.

$\longmapsto \sf { \dfrac{ 1}{x} = \dfrac{1+ \sqrt{2} }{1 - 2}}$

Performing subtraction.

$\longmapsto \sf { \dfrac{ 1}{x} = \dfrac{1+ \sqrt{2} }{-1}}$

Multiplying the numerator and the denominator with -1 in order to make the denominator positive.

$\longmapsto \sf { \dfrac{ 1}{x} = \dfrac{-1(1+ \sqrt{2} )}{(-1)(-1)}}$

Performing multiplication.

$\longmapsto \sf { \dfrac{ 1}{x} = \dfrac{-1- \sqrt{2} }{1}}$

Now, we can write it as,

$\longmapsto \bf { \dfrac{ 1}{x} = -1- \sqrt{2}}$

Now, substitute the values in the expression we have to find out as per the question given.

$\longmapsto \sf { { \Bigg ( x - \dfrac{1}{x} \Bigg )}^{2} }$

Substituting values.

$\longmapsto \sf {{ \Big \{ 1-\sqrt{2}- (-1- \sqrt{2}) \Big \} }^{2} }$

Removing the brackets.

$\longmapsto \sf {{ \Big \{ 1-\sqrt{2}+1 +\sqrt{2}) \Big \} }^{2} }$

Performing subtraction. +√2 and -√2 will cancel out.

$\longmapsto \sf { (2)^{2} }$

$\longmapsto \bf{ 4 }$

∴ 4 is the required answer.