Question

Rationalised the denominator of 1/root5-root2-root7

Answer 1

$\frac{1}{(\sqrt{5} - \sqrt{2} - \sqrt{7})} \\= \frac{(\sqrt{5} - \sqrt{2} + \sqrt{7})}{(\sqrt{5} - \sqrt{2} - \sqrt{7}) (\sqrt{5} - \sqrt{2} + \sqrt{7})}$

$= \frac{(\sqrt{5} - \sqrt{2} + \sqrt{7})}{(\sqrt{5} - \sqrt{2} )^{2} - (\sqrt{7})^{2} }$

$= \frac{(\sqrt{5} - \sqrt{2} + \sqrt{7})}{(\sqrt{5})^{2}+ (\sqrt{2} )^{2} - 2\times \sqrt{5} \times \sqrt{2} - (\sqrt{7})^{2} }$

$= \frac{(\sqrt{5} - \sqrt{2} + \sqrt{7})}{5+ 2 - 2\times \sqrt{10}- 7 }$

$= \frac{\sqrt{10}(\sqrt{5} - \sqrt{2} + \sqrt{7})}{(- 2\times \sqrt{10} ) (\sqrt{10})}$

$= \frac{(5\sqrt{2} - 2\sqrt{5} + \sqrt{70})} { -200}$

Therefore.,

$\red{ \frac{1}{(\sqrt{5} - \sqrt{2} - \sqrt{7})}}$

$\green {= - \frac{(5\sqrt{2} - 2\sqrt{5} + \sqrt{70})} { 200}}$

•••♪

Answer 2

Answer:

√70-5√2+2√5/20

Step-by-step explanation:

1/√7+√5-√2

=1(√7)+(√5-√2)/(√7)+(√5-√2)*(√7)+(√5-√2)

=√7-(√5-√2)/(√7)²-(√5-√2)² {(a+b)(a-b)=a^2-b^2}

=√7-√5+√2/7-{(√5)²+(√2)²-2(√5)(√2)}  {(a-b)²=a²+b²-2ab}

=√7-√5+2/7-[5+2-2√10]

=√7-√5+2/7-5-2+2√10

=√7-√5+2/2√10

=√7-√5+2*√10/2√10*√10

=(√7-√5+2)√10/2*√10*√10

=√70-√50+√20/2*10

=√70-5√2+2√5/20

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