Question

Rationalise the following √2-1/√2+1

Answer 1

Calculation:

The given fraction is an irrational one. To rationalize it, we have to multiply its numerator and denominator by a factor which will give a rational product with a rational denominator. Towards this end, we multiply the numerator and the denominator of the given expression by √(√2 - 1) and write,

√(√ 2 - 1)/√(√ 2 + 1) = √(√ 2 - 1)/√(√ 2 + 1) x √(√ 2 - 1)/√(√ 2 - 1)

= [√(√ 2 - 1) x √(√ 2 - 1)] / [√(√ 2 + 1) √(√ 2 - 1)] = (√ 2 - 1) / √[(√ 2 + 1)(√ 2 - 1)]

=(√ 2 - 1) / √[(√2)² - 1² )] = (√2 - 1)/√(2 - 1) = (√2 - 1)/√1 = (√2 - 1)/1 = √2 - 1 (Considering only the positive root).

= 1.4142 - 1 = 0.4142, to 4 decimal figures

Answer 2

☯ SolutioN :

As, for rationalizing the denominator firstly we have to learn the given below steps ⤵⤵⤵

1. Firstly, for rationalizing the denominator the number must be irrational.
2. After, that we will subtract or add the like numbers.
3. When there would be unlike numbers then we will multiply the number which is in denominator by both numerator and denominator.
4. While multiplying the number BT both numerator and denominator we will change the last sign of denominator. But if only one number is left then we will not change the sign.
5. After, that we will multiply them.

$\rule{200}{1}$

Now,

We will Rationalize the following question

$\sf{\dashrightarrow \frac{\sqrt{2} - 1}{\sqrt{2} + 1}} \\ \\ \sf{\dashrightarrow \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1}} \\ \\ \sf{\dashrightarrow \frac{2 - \sqrt{2} - \sqrt{2} + 1}{(\sqrt{2})^2 - (1)^2}} \\ \\ \sf{\dashrightarrow \frac{3 - 2 \sqrt{2}}{1}} \\ \\ \Large{\implies{\boxed{\boxed{\sf{3 - 2\sqrt{2}}}}}}$

$\therefore$ 3 - 2√2 is required answer.