Question

if x = 1 -√2, find the value of [ x -1/x] ^3​

Answer 1

Answer:

The value of (x - 1/x)³ is 8

Explanation:

x = 1 - √2

First, let's find the value of 1/x.

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

We have to rationalize the denominator.

• The factor of multiplication by which rationalization is done, is called as rationalizing factor.

• To find the rationalizing factor,

=>  If the denominator contains 2 terms, just change the sign between the two terms.  

For example, rationalizing factor of (3 + √2) is (3 - √2)

So, rationalizing factor of (1 - √2) = (1 + √2)

Now, multiply the numerator and denominator of 1/x by the rationalizing factor.

$\tt \dfrac{1}{x}=\dfrac{1}{1-\sqrt{2}} \times \dfrac{1+\sqrt{2}}{1+\sqrt{2}} \\\\ \tt \dfrac{1}{x}=\dfrac{1+\sqrt{2}}{(1-\sqrt{2})(1+\sqrt{2})}$

Applying the identity (a - b) (a + b) = a² - b²

$\tt \dfrac{1}{x}=\dfrac{1+\sqrt{2}}{1^2-\sqrt{2}^2} \\\\ \tt \dfrac{1}{x}=\dfrac{1+\sqrt{2}}{1-2} \\\\ \tt \dfrac{1}{x}=\dfrac{1+\sqrt{2}}{-1} \\\\ \tt \dfrac{1}{x}=-(1+\sqrt{2}) \\\\ \longrightarrow \boxed{\tt \dfrac{1}{x}=-1-\sqrt{2}}$

Finding the value of (x - 1/x)³

$\mapsto \sf \bigg(x-\dfrac{1}{x} \bigg)^3 \\\\ \mapsto \sf (1-\sqrt{2}-[-1-\sqrt{2}])^3 \\\\ \mapsto \sf (1-\sqrt{2}+1+\sqrt{2})^3 \\\\ \mapsto \sf 2^3 \\\\ \mapsto \sf 8$