Question

please solve this ques step by step​

Answer 1

QUESTION:

$\frac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} } -\frac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{5} } - \frac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} }$

TO DO:

SIMPLIFICATION.

EXPLANATION:

RATIONALZING THE FIRST TERM:

$IDENTITY\:USED:\\ (A-B)(A+B)=(A+B)^2$

$\frac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} } \times \frac{ \sqrt{10} - \sqrt{3} }{ \sqrt{10} - \sqrt{3} } \\ = \frac{7 \sqrt{3} ( \sqrt{10} - \sqrt{3} ) }{ { (\sqrt{10}) }^{2} - {( \sqrt{3} )}^{2} } \\ = \frac{7 \sqrt{3} ( \sqrt{10} - \sqrt{3} )}{10 - 3} \\ = \frac{7 \sqrt{3}( \sqrt{10} - \sqrt{3} ) }{7} \\ = \sqrt{30} - 3 </p><p>$

RATIONALZING THE SECOND TERM:

$\frac{ - 2 \sqrt{5} }{ \sqrt{6} + \sqrt{5} } \times \frac{ \sqrt{6} - \sqrt{5} }{ \sqrt{6} - \sqrt{5} } \\ = \frac{ - 2 \sqrt{5} ( \sqrt{6} - \sqrt{5}) }{ { (\sqrt{6}) }^{2} - {( \sqrt{5)} }^{2} } \\ = \frac{ - 2 \sqrt{30} + 2(5)}{6 - 5} \\ = 10 - 2 \sqrt{30}$

RATIONALIZING THE THIRD TERM:

$\frac{ - 3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } \times \frac{\sqrt{15} - 3 \sqrt{2}}{\sqrt{15} - 3 \sqrt{2}}\\ = \frac{ - 3 \sqrt{2}(\sqrt{15} - 3 \sqrt{2})}{ {( \sqrt{15)} }^{2} - {(3 \sqrt{2)} }^{2} } \\ = \frac{ - 3( \sqrt{30 } - 6) }{15 - (9 \times 2)} \\ = \frac{ - 3( \sqrt{30} - 6) }{ - 3} \\ = \sqrt{30} - 6$

ADD THE TERMS AS WE HAVE TOOK THE TERM WITH NEGATIVE SIGN:

$\sqrt{30} - 3+10 - 2 \sqrt{30}+\sqrt{30} - 6\\ = - 9 + 10 + 2 \sqrt{30} - 2 \sqrt{30} \\ = 10 - 9 \\ = 1$

HENCE ANSWER IS 1.