Question

Rationalise the denominator of the following :- 1/3+√2

Answer 1

Answer:

The required fraction with rationalised denominator is

$\boxed{\red{\sf\:\dfrac{3\:-\:\sqrt{2}}{7}}}$

Step-by-step explanation:

The given fraction with irrational denominator is

$\sf\dfrac{1}{3\:+\:\sqrt{2}}$.

We have to rationalise the denominator of this fraction.

To rationalise the denominator, we have to multiply both numerator and denominator by the conjugate pair of the denominator.

$\therefore\sf\:\dfrac{1}{3\:+\:\sqrt{2}}\\\\\implies\sf\:\dfrac{1}{3\:+\:\sqrt{2}}\:\times\:\dfrac{3\:-\:\sqrt{2}}{3\:-\:\sqrt{2}}\\\\\implies\sf\:\dfrac{3\:-\:\sqrt{2}}{(\:3\:)^{2}\:-\:(\:\sqrt{2}\:)^{2}}\:\:\:-\:-\:-\:[\:(\:a\:+\:b\:)\:(\:a\:-\:b\:)\:=\:a^{2}\:-\:b^{2}\:]\\\\\implies\sf\:\dfrac{3\:-\:\sqrt{2}}{9\:-\:2}\\\\\implies\boxed{\red{\sf\:\dfrac{3\:-\:\sqrt{2}}{7}}}$

Additional Information:

1. Fraction:

Any rational number in the form of $\sf\:\dfrac{p}{q}$ is called as a fraction.

2. Types of fraction:

There are three types of fraction:

A. Proper fraction

B. Improper fraction

C. Mixed fraction

3. Proper fraction:

In this type of fraction, the numerator is less than the denominator.

4. Improper fraction:

In this type of fraction, the denominator is less than the numerator.

In this type of fraction, numerator is also equal to denominator sometimes.

5. Mixed fraction:

In this type of fraction, there is a whole number with a proper fraction.

6. Rationalised denominator:

When the denominator of a fraction is a rational number, then the denominator of the fraction is called rationalised denominator.

7. Rationalisation of denominator:

To rationalise the denominator of a fraction, we have to multiply both numerator and denominator by the conjugate pair of denominator.

8. Conjugate pair:

If the given number is sum of two numbers, then its conjugate pair is subtraction of those two numbers.

For example,

The conjugate pair of $\boxed{\red{\sf{4\:+\:\sqrt{5}}}}$ is $\boxed{\red{\sf{4\:-\:\sqrt{5}}}}$.