Question

rationalise the denominator of 2√3/7-2√3​

Answer 1

We have been asked to rationalise the denominator of $\sf{\dfrac{2\sqrt{3}}{7 - 2\sqrt{3}}}$.

For this we need to follow some distinct steps listed down :

1. We need to check out the symbol in the given rational number. Then change its form and multiply the same with both the numerator as well as the denominator.
2. After doing so we will get both the denominator is in the form of (a+b) (a - b) identity.
3. Following the extension of the Identity that is a² - b² square. Following we will simplify the denominator to rationalise the denominator.

Solution :

$\small{\longrightarrow{\sf{\dfrac{2 \sqrt{3}}{7 - 2\sqrt{3}}}}}$

$\small{\longrightarrow{\sf{\dfrac{2 \sqrt{3}}{7 - 2\sqrt{3}} \times \dfrac{7 + 2\sqrt{3}}{7 + 2\sqrt{3}}}}}$

$\small{\longrightarrow{\sf{\dfrac{ 7( 2\sqrt{3}) \times 2\sqrt{3}(2 \sqrt{3})}{{(7)}^{2} - {(2 - \sqrt{3}}^{2})}}}}$

$\small{\longrightarrow{\sf{\dfrac{7( 2 \sqrt{3}) \times 2\sqrt{3}(2 \sqrt{3})}{(49) - (2 \times 2)(\sqrt{3} \times \sqrt{3})}}}}$

$\small{\longrightarrow{\sf{\dfrac{7( 2\sqrt{3}) \times 2\sqrt{3}(2 \sqrt{3})}{49 - 4 \times 3}}}}$

$\small{\longrightarrow{\sf{\dfrac{ 7( 2\sqrt{3}) \times 2\sqrt{3}(2 \sqrt{3})}{49 - 4 \times 3}}}}$

$\small{\longrightarrow{\sf{\dfrac{7( 2\sqrt{3}) \times 2\sqrt{3}(2 \sqrt{3})}{49 - 12}}}}$

$\large{\boxed{\implies{\sf \pink{\dfrac{14\sqrt{3} + 12}{37}}}}}$

Hence rationalized .

Answer 2

Step-by-step explanation:

$\large\bf Given:-$

• $\large\sf \dfrac{2 \sqrt{3} }{7 - 2 \sqrt{3} }$

We have to rationalise the denominator.

Solution:-

$\implies\sf \dfrac{2 \sqrt{3} }{7 - 2 \sqrt{3} } \times \dfrac{7 + 2 \sqrt{3} }{7 + 2 \sqrt{3} }$

$\implies\sf \dfrac{(2 \sqrt{3}) \times (7 + 2 \sqrt{3}) }{(7 - 2 \sqrt{3} ) \times (7 + 2 \sqrt{3})}$

$\implies\sf \dfrac{7(2 \sqrt{3}) +2 \sqrt{3} (2 \sqrt{3}) }{(7 - 2 \sqrt{3} ) (7 + 2 \sqrt{3})}$

As we know that (a + b)(a -b ) = a² - b²

$\implies\sf \dfrac{7(2 \sqrt{3}) +2 \sqrt{3} \times 2 \sqrt{3}}{{7}^{2} - {(2 \sqrt{3}) }^{2}}$

$\implies\sf \dfrac{7(2 \sqrt{3}) +2\times 2 \times\sqrt{3} \times \sqrt{3}}{{7}^{2} - {(2 \sqrt{3}) }^{2}}$

$\implies\sf \dfrac{14\sqrt{3} +12} { {7}^{2} - {(2 \sqrt{3}) }^{2} }$

$\implies\sf \dfrac{14\sqrt{3} +12} { 49 - 12 }$

$\implies\sf \dfrac{14\sqrt{3} +12} { 37 }$

$\boxed{ \purple{\rightarrow\sf \dfrac{14\sqrt{3} +12} { 37 }} }$