Question

Rationalise :
$\dashrightarrow \tt \boxed{ \green{ \cfrac{3 + 2 \sqrt{5} }{5 - 2 \sqrt{3} } }}$
Answer It As Soon As Possible !!​

Answer 1

$\large{ \sf \underline{Solution - }}$

To rationalise :

$\sf \implies\cfrac{3 + 2 \sqrt{5} }{5 - 2 \sqrt{3} }$

➢ Here the denominator is in the form of (a-b). Rationalising factor of (a-b) is (a+b). So, rationalising factor of (5-2√3) is (5+2√3). We will multiply (5+2√3) with both the numerator and denominator to rationalise the denominator.

Now,

$\sf \implies\cfrac{3 + 2 \sqrt{5} }{5 - 2 \sqrt{3} } \times\cfrac{5+2 \sqrt{3} }{5+2 \sqrt{3} }$

Combine fractions

${ \sf \implies\cfrac{(3 + 2 \sqrt{5})(5+2 \sqrt{3} ) }{(5 - 2 \sqrt{3})({5+2 \sqrt{3}) } }}$

We know that,

$\sf \implies (a + b)(a - b) = a^{2} - b ^{2}$

So,

$\sf \implies\cfrac{(3 + 2 \sqrt{5})(5+2 \sqrt{3} ) }{(5) ^{2} - (2 \sqrt{3})^{2} }$

Simplify the denominator,

$\sf \implies\cfrac{(3 + 2 \sqrt{5})(5+2 \sqrt{3} ) }{25 - 12 }$

$\sf \implies\cfrac{(3 + 2 \sqrt{5})(5+2 \sqrt{3} ) }{13 }$

Now, simplify the numerator,

$\sf \implies\cfrac{3(5 + 2 \sqrt{3}) + 2 \sqrt{5} (5 + 2 \sqrt{3} ) }{13 }$

$\sf \implies\cfrac{15 + 6 \sqrt{3} + 10 \sqrt{5} + 4 \sqrt{15} }{13 }$

Hence,

On rationalising we got,

$\bf \implies\cfrac{15 + 6 \sqrt{3} + 10 \sqrt{5} +4\sqrt{15} }{13 }$