Question

Rationalise the denominator 1/3 + √7

Answer 1

Given :–

• A fraction: $\dfrac{1}{3 + \sqrt{7}}$

Required :–

• Rationalise the denominator of the given fraction.

Solution :–

First, we need to multiply the additive Inverse of the denominator with the given fraction.

• Additive Inverse = change the sign.

Example :– positive (+ve) number becomes negative (–ve) or negative (–ve) number becomes positive (+ve).

So, the additive Inverse of 3 + √7 is 3 – √7.

Now, multiply the additive Inverse by the given fraction,

⟹ $\dfrac{1}{3 + \sqrt{7}} \times \dfrac{ 3 - \sqrt {7}}{3 - \sqrt{7}}$

We know that,

(a + b)(a – b) = a² – b²

So,

⟹ $\dfrac{3 - \sqrt{7}}{{(3)}^{2} - {(\sqrt{7})}^{2}}$

Powers cuts the square roots, it means

⟹ $\dfrac{3 - \sqrt{7}}{9 - 7}$

⟹ $\dfrac{3 - \sqrt{7}}{2}$

Hence,

The rationalise denominator of $\dfrac{1}{3 + \sqrt{7}}$ is 2.

Answer 2

(3 - √7)/2

Step-by-step explanation:

$\bf \underline{Solution-}$

Given expression

1/(3 + √7)

The denominator = 3 + √7.

We know that

Rationalising factor of a + √b = a - √b.

So, the rationalising factor of 3 + √7 = 3 - √7.

On rationalising the denominator them

→ [1/(3 + √7)] × [(3 - √7)/(3 - √7)]

→ [1(3 - √7)]/[(3 + √7)(3 - √7)]

{.°. (a +b)(a-b) = a² - b²}

→ (3 - √7)/[(3)² - (√7)²]

→ (3 - √7)/(9 - 7)

→ (3 - √7)/2

Hence, the denominator is rationalised.