Simplify-i 4-24-2ii 5-35-3iii 6-66-6iv 5-22-3v 5-32vi 3-32

(i) #160;4+24 2=42 22 #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160;[a+ba b=a2 b2]=16 2=14 #160; #160; ...

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Byju's Answer

Standard IX

Mathematics

Conditional Identities

Simplify:i 4+...

Question

Simplify:
(i) $(4 + \sqrt{2}) (4 - \sqrt{2})$
(ii) $(\sqrt{5} + \sqrt{3}) (\sqrt{5} - \sqrt{3})$
(iii) $(6 - \sqrt{6}) (6 + \sqrt{6})$
(iv) $(\sqrt{5} - \sqrt{2}) (\sqrt{2} - \sqrt{3})$
(v) ${(\sqrt{5} - \sqrt{3})}^{2}$
(vi) ${(3 - \sqrt{3})}^{2}$

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Solution

$(i) (4 + \sqrt{2}) (4 - \sqrt{2}) = 4^{2} - {(\sqrt{2})}^{2} [(a + b) (a - b) = a^{2} - b^{2}] = 16 - 2 = 14 (ii) (\sqrt{5} + \sqrt{3}) (\sqrt{5} - \sqrt{3}) = {(\sqrt{5})}^{2} - {(\sqrt{3})}^{2} = 5 - 3 = 2 (iii) (6 - \sqrt{6}) (6 + \sqrt{6}) = 6^{2} - {(\sqrt{6})}^{2} = 36 - 6 = 30 (iv) (\sqrt{5} - \sqrt{2}) (\sqrt{2} - \sqrt{3}) = \sqrt{5} \times \sqrt{2} - \sqrt{5} \times \sqrt{3} - \sqrt{2} \times \sqrt{2} + \sqrt{2} \times \sqrt{3} = \sqrt{10} - \sqrt{15} - 2 + \sqrt{6} (v) {(\sqrt{5} - \sqrt{3})}^{2} = {(\sqrt{5})}^{2} + {(\sqrt{3})}^{2} - 2 \times \sqrt{5} \sqrt{3} [{(a - b)}^{2} = a^{2} + b^{2} - 2 a b] = 5 + 3 - 2 \sqrt{15} = 8 - 2 \sqrt{15} (vi) {(3 - \sqrt{3})}^{2} = 3^{2} + {(\sqrt{3})}^{2} - 2 \times 3 \times \sqrt{3} [{(a - b)}^{2} = a^{2} + b^{2} - 2 a b] = 9 + 3 - 6 \sqrt{3} = 12 - 6 \sqrt{3}$

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Similar questions

Q. Simplify:

(i)

\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} + \frac{\sqrt{12}}{\sqrt{3} - \sqrt{2}}

(ii)

\frac{5 + \sqrt{3}}{5 - \sqrt{3}} + \frac{5 - \sqrt{3}}{5 + \sqrt{3}}

(iii)

\frac{7 + 3 \sqrt{5}}{3 + \sqrt{5}} + \frac{7 - 3 \sqrt{5}}{3 - \sqrt{5}}

(iv)

\frac{1}{2 + \sqrt{3}} + \frac{2}{\sqrt{5} - \sqrt{3}} + \frac{1}{2 - \sqrt{5}}

(v)

\frac{2}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{3}{\sqrt{5} + \sqrt{2}}

Simplify:

(i) $(16^{\frac{- 1}{5}})^{\frac{5}{2}}$

(ii) $\sqrt[5]{(32)^{- 3}}$

(iii) $\sqrt[3]{(343)^{- 2}}$

(iv) $(0.001)^{\frac{1}{3}}$

(v) $\frac{(25)^{\frac{3}{2}} \times (243)^{\frac{3}{5}}}{(16)^{\frac{5}{4}} \times (8)^{\frac{4}{3}}}$

(vi) ${(\frac{\sqrt{2}}{5})}^{8} \div {(\frac{\sqrt{2}}{5})}^{13}$

(vii) ${(\frac{5^{- 1} \times 7^{2}}{5^{2} \times 7^{- 4}})}^{\frac{7}{2}} \times {(\frac{5^{- 2} \times 7^{3}}{5^{3} \times 7^{- 5}})}^{\frac{- 5}{2}}$

Q. Simplify:

(i)

(16^{- 1 / 5})^{5 / 2}

(ii)

\sqrt[5]{{(32)}^{- 3}}

(iii)

\sqrt[3]{(343)^{- 2}}

(iv)

(0.001)^{1 / 3}

(v)

\frac{(25)^{3 / 2} \times (243)^{3 / 5}}{(16)^{5 / 4} \times (8)^{4 / 3}}

(vi)

{(\frac{\sqrt{2}}{5})}^{8} \div {(\frac{\sqrt{2}}{5})}^{13}

(vii)

{(\frac{5^{- 1} \times 7^{2}}{5^{2} \times 7^{- 4}})}^{7 / 2} \times {(\frac{5^{- 2} \times 7^{3}}{5^{3} \times 7^{- 5}})}^{- 5 / 2}

Q. Simplify:

(i)

\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} + \frac{\sqrt{12}}{\sqrt{3} - \sqrt{2}}

(ii)

\frac{7 + 3 \sqrt{5}}{3 + \sqrt{5}} - \frac{7 - 3 \sqrt{5}}{3 - \sqrt{5}}

Q. Simplify:
(i)

(3^{2} + 2^{2}) \times {(\frac{1}{2})}^{3}

(ii)

(3^{2} - 2^{2}) \times {(\frac{2}{3})}^{- 3}

(iii)

[{(\frac{1}{3})}^{- 3} - {(\frac{1}{2})}^{- 3}] \div {(\frac{1}{4})}^{- 3}

(iv)

(2^{2} + 3^{2} - 4^{2}) \div {(\frac{3}{2})}^{2}

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Simplify: (i) 4+24-2 (ii) 5+35-3 (iii) 6-66+6 (iv) 5-22-3 (v) 5-32 (vi) 3-32

(i) 4+24-2=42-22 [a+ba-b=a2-b2]=16-2=14 (ii) 5+35-3=52-32=5-3=2(iii) 6-66+6=62-62=36-6=30(iv) 5-22-3=5×2-5×3-2×2+2×3 =10-15-2+6(v) 5-32=52+32-2×53 [a-b2=a2+b2-2ab]=5+3-215=8-215(vi) 3-32=32+32-2×3×3 [a-b2=a2+b2-2ab]=9+3-63=12-63

Simplify:
(i) $(4 + \sqrt{2}) (4 - \sqrt{2})$
(ii) $(\sqrt{5} + \sqrt{3}) (\sqrt{5} - \sqrt{3})$
(iii) $(6 - \sqrt{6}) (6 + \sqrt{6})$
(iv) $(\sqrt{5} - \sqrt{2}) (\sqrt{2} - \sqrt{3})$
(v) ${(\sqrt{5} - \sqrt{3})}^{2}$
(vi) ${(3 - \sqrt{3})}^{2}$