Simplifyi 4-54-5-4-54-5ii 13-2-25-3-32-5iii 2-32-3-2-32-3-3-13-1iv 262-3-626-3-836-2

(i) 4+54 5+4 54+5=4+54 5 #215;4+54+5+4 54+5 #215;4 54 5=4+5242 52+4 5242 52 =16+5+85+16+5 8516 5=4211 (ii) 13+2 25 3 32 5=13+2 #215;3 23 2 25 3 #215;5+3 ...

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Byju's Answer

Standard IX

Mathematics

Conditional Identities

Simplifyi 4+5...

Question

Simplify

(i) $\frac{4 + \sqrt{5}}{4 - \sqrt{5}} + \frac{4 - \sqrt{5}}{4 + \sqrt{5}}$

(ii) $\frac{1}{\sqrt{3} + \sqrt{2}} - \frac{2}{\sqrt{5} - \sqrt{3}} - \frac{3}{\sqrt{2} - \sqrt{5}}$

(iii) $\frac{2 + \sqrt{3}}{2 - \sqrt{3}} + \frac{2 - \sqrt{3}}{2 + \sqrt{3}} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$

(iv) $\frac{2 \sqrt{6}}{\sqrt{2} + \sqrt{3}} + \frac{6 \sqrt{2}}{\sqrt{6} + \sqrt{3}} - \frac{8 \sqrt{3}}{\sqrt{6} + \sqrt{2}}$

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Solution

(i)
$\frac{4 + \sqrt{5}}{4 - \sqrt{5}} + \frac{4 - \sqrt{5}}{4 + \sqrt{5}} = \frac{4 + \sqrt{5}}{4 - \sqrt{5}} \times \frac{4 + \sqrt{5}}{4 + \sqrt{5}} + \frac{4 - \sqrt{5}}{4 + \sqrt{5}} \times \frac{4 - \sqrt{5}}{4 - \sqrt{5}} = \frac{{(4 + \sqrt{5})}^{2}}{{(4)}^{2} - {(\sqrt{5})}^{2}} + \frac{{(4 - \sqrt{5})}^{2}}{{(4)}^{2} - {(\sqrt{5})}^{2}}$
$= \frac{16 + 5 + 8 \sqrt{5} + 16 + 5 - 8 \sqrt{5}}{16 - 5} = \frac{42}{11}$
(ii)
$\frac{1}{\sqrt{3} + \sqrt{2}} - \frac{2}{\sqrt{5} - \sqrt{3}} - \frac{3}{\sqrt{2} - \sqrt{5}} = \frac{1}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} - \frac{2}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} - \frac{3}{\sqrt{2} - \sqrt{5}} \times \frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} + \sqrt{5}} = \frac{\sqrt{3} - \sqrt{2}}{{(\sqrt{3})}^{2} - {(\sqrt{2})}^{2}} - \frac{2 (\sqrt{5} + \sqrt{3})}{{(\sqrt{5})}^{2} - {(\sqrt{3})}^{2}} - \frac{3 (\sqrt{2} + \sqrt{5})}{{(\sqrt{2})}^{2} - {(\sqrt{5})}^{2}}$
$= \frac{\sqrt{3} - \sqrt{2}}{3 - 2} - \frac{2 (\sqrt{5} + \sqrt{3})}{5 - 3} - \frac{3 (\sqrt{2} + \sqrt{5})}{2 - 5} = \sqrt{3} - \sqrt{2} - \frac{2 (\sqrt{5} + \sqrt{3})}{2} - \frac{3 (\sqrt{2} + \sqrt{5})}{(- 3)} = \sqrt{3} - \sqrt{2} - \sqrt{5} - \sqrt{3} + \sqrt{2} + \sqrt{5}$
$= 0$
(iii)
$\frac{2 + \sqrt{3}}{2 - \sqrt{3}} + \frac{2 - \sqrt{3}}{2 + \sqrt{3}} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} + \frac{2 - \sqrt{3}}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \times \frac{\sqrt{3} - 1}{\sqrt{3} - 1} = \frac{{(2 + \sqrt{3})}^{2}}{{(2)}^{2} - {(\sqrt{3})}^{2}} + \frac{{(2 - \sqrt{3})}^{2}}{{(2)}^{2} - {(\sqrt{3})}^{2}} + \frac{{(\sqrt{3} - 1)}^{2}}{{(\sqrt{3})}^{2} - 1^{2}}$
$= \frac{4 + 3 + 4 \sqrt{3}}{4 - 3} + \frac{4 + 3 - 4 \sqrt{3}}{4 - 3} + \frac{3 + 1 - 2 \sqrt{3}}{3 - 1} = 7 + 4 \sqrt{3} + 7 - 4 \sqrt{3} + \frac{4 - 2 \sqrt{3}}{2} = 14 + 2 - \sqrt{3}$
$= 16 - \sqrt{3}$
(iv)
$\frac{2 \sqrt{6}}{\sqrt{2} + \sqrt{3}} + \frac{6 \sqrt{2}}{\sqrt{6} + \sqrt{3}} - \frac{8 \sqrt{3}}{\sqrt{6} + \sqrt{2}} = \frac{2 \sqrt{6}}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} + \frac{6 \sqrt{2}}{\sqrt{6} + \sqrt{3}} \times \frac{\sqrt{6} - \sqrt{3}}{\sqrt{6} - \sqrt{3}} - \frac{8 \sqrt{3}}{\sqrt{6} + \sqrt{2}} \times \frac{\sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}} = \frac{2 \sqrt{6} \times \sqrt{3} - 2 \sqrt{6} \times \sqrt{2}}{{(\sqrt{3})}^{2} - {(\sqrt{2})}^{2}} + \frac{6 \sqrt{2} \times \sqrt{6} - 6 \sqrt{2} \times \sqrt{3}}{{(\sqrt{6})}^{2} - {(\sqrt{3})}^{2}} - \frac{8 \sqrt{3} \times \sqrt{6} - 8 \sqrt{3} \times \sqrt{2}}{{(\sqrt{6})}^{2} - {(\sqrt{2})}^{2}}$
$= \frac{2 \sqrt{18} - 2 \sqrt{12}}{3 - 2} + \frac{6 \sqrt{12} - 6 \sqrt{6}}{6 - 3} - \frac{8 \sqrt{18} - 8 \sqrt{6}}{6 - 2} = 2 \sqrt{18} - 2 \sqrt{12} + \frac{6 \sqrt{12} - 6 \sqrt{6}}{3} - \frac{8 \sqrt{18} - 8 \sqrt{6}}{4} = 2 \sqrt{18} - 2 \sqrt{12} + 2 \sqrt{12} - 2 \sqrt{6} - 2 \sqrt{18} + 2 \sqrt{6}$
$= 0$

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Simplify (i) 4+54-5+4-54+5 (ii) 13+2-25-3-32-5 (iii) 2+32-3+2-32+3+3-13+1 (iv) 262+3+626+3-836+2