Question

Rationalise the denominator of
of
$\frac{3 \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} }$
​

Answer 1

Step-by-step explanation:

$= = > \frac{3 \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \\ = = > \frac{3 \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{ 3} }{ \sqrt{5} + \sqrt{3} } \\ = = > \: \frac{(3 \sqrt{5} + \sqrt{3}) \times ( \sqrt{5} + \sqrt{ 3} )}{( \sqrt{5}) {}^{2} - ( \sqrt{3}) {}^{2} } \\ \frac{3 \sqrt{5} \times \sqrt{5} + 3 \sqrt{5} \times \sqrt{3} + \sqrt{3} \times \sqrt{5} + \sqrt{3} \times \sqrt{3} }{5 - 3} \\ = = > \frac{15 + 3 \sqrt{15} + \sqrt{15} + 3 }{2} \\ = = > \frac{18 + 4 \sqrt{15} }{2} \\ = = > \frac{2(9 + 2 \sqrt{15}) }{2} \\ = = > 9 + 2 \sqrt{15}$

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SOME TIPS TO SOLVE THESE TYPE OF QUESTIONS EASILY:-

a) PRECISE CALCULATION IS MUST A SMALL ERROR IN CALCULATION CAN MAKE YOUR WHOLE ANSWER WRONG.

b) YOU NEED TO KNOW CERTAIN FORMULAS TO SOLVE THESE TYPE OF QUESTIONS FOR EXAMPLE:-

$= = > a {}^{2} - b {}^{2} = (a + b) \times (a - b)$

c) YOU NEED TO KNOW METHOD TO RATIONALISE A FRACTION:-

1) JUST YOU NEED TO MULTIPLE THE FRACTION WITH SAME NUMBER IN NUMERATOR AND DENOMINATOR WHICH OPPOSITE SIGN OF THE DENOMINATORS SIGN.

$\underline\green{ \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times }$

Answer 2

⠀⠀⠀⠀${ \huge {\mathtt{ \blue{QUESTION}}}} </p><p>$

Rationalise the denominator of

of ⠀⠀⠀⠀${\bf \frac{3 \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } }$

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⠀⠀⠀⠀⠀$\huge\underline{ \underline{ \orange{ \bold{sOluTiOn}}}}$

⠀⠀⠀⠀⠀

Multiply the fraction by $\bf\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

➩⠀⠀⠀⠀⠀$\bf\:\frac{3\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$

➩⠀⠀⠀⠀⠀$\bf \frac{3 \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} }$

To multiply the fraction , multiply the numerators and denominators separately

➩⠀⠀⠀⠀⠀$\bf \frac{(3 \sqrt{5} + \sqrt{3}) \times ( \sqrt{5} + \sqrt{3}) }{ (\sqrt{5} - \sqrt{3}) \times \sqrt{5} + \sqrt{ 3} }$

➩⠀⠀⠀$\bf \frac{15 + 3 \sqrt{15} + \sqrt{15} + 3}{( \sqrt{5} - \sqrt{3} )× ( \sqrt{5} + \sqrt{ 3}) }$

⠀⠀

using$\:\underline{\boxed{\green{\bf{(a-b)(a+b)=a{}^{2}-{b}^{2}}}}}$⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀simplify the product.

➩⠀⠀⠀$\bf\:\frac{18+4\sqrt{15}}{2}$⠀

⠀⠀⠀ Factor out the expression ⠀

➩⠀⠀⠀⠀⠀$\bf\:\frac{2(9+2\sqrt{15})}{2}$

⠀⠀⠀⠀Reduce the fraction with 2

➩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\:{\underline{\boxed{\red{\bf{9+2\sqrt{15}}}}}}$

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