Simplify- 1-256-1-3 - 1-256-1-2 - 2-243-2-3Prove that - 1-1- radic-2- - 1- radic-2- radic-3- - 1- radic-2- radic-3- - 1- radic-2- radic-3- - 1- radic-2- radic-3- - 1- radic-2- radic-3 - 1- radic-2- radic-3 - 1- radic-2- radic-3- - 2 -Prove that- 81-16-3-2 x - 25-9-3-2 - 5-2-1 - 1

1) #160;125613+125612+224323= #160;126 #215;2213+12812+233 #215;3223= #160;126 #215;13 #215;223+124+233 #215;23 #215;32 #215;23= #160;14 ...

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Byju's Answer

Standard IX

Mathematics

Properties of Multiplication of Integers

Simplify: 1/(...

Question

Simplify: 1/(256)1/3 + 1/(256)1/2 + 2/(243)2/3
Prove that : 1/(1+ √2) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3 )
+ 1/(√2+ √3 )+ 1/(√2+ √3) = 2
.Prove that: {81/16}-3/2 x [ {25/9}-3/2 + {5/2}-1 = 1

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Solution

$1) \frac{1}{{(256)}^{\frac{1}{3}}} + \frac{1}{{(256)}^{\frac{1}{2}}} + \frac{2}{{(243)}^{\frac{2}{3}}} = \frac{1}{{(2^{6} \times 2^{2})}^{\frac{1}{3}}} + \frac{1}{{(2^{8})}^{\frac{1}{2}}} + \frac{2}{{(3^{3} \times 3^{2})}^{\frac{2}{3}}} = \frac{1}{2^{6 \times \frac{1}{3}} \times 2^{\frac{2}{3}}} + \frac{1}{2^{4}} + \frac{2}{3^{3 \times \frac{2}{3}} \times 3^{2 \times \frac{2}{3}}} = \frac{1}{4 \times 2^{\frac{2}{3}}} + \frac{1}{16} + \frac{2}{9 \times 3^{\frac{4}{3}}} = \frac{1}{4 \times 2^{\frac{2}{3}}} \times \frac{2^{\frac{1}{3}}}{2^{\frac{1}{3}}} + \frac{1}{16} + \frac{2}{9 \times 3^{\frac{4}{3}}} \times \frac{3^{\frac{2}{3}}}{3^{\frac{2}{3}}} {rationalising the denominator of first and third term} = \frac{2^{\frac{1}{3}}}{8} + \frac{1}{16} + \frac{2 \times 3^{\frac{2}{3}}}{81} = \frac{\sqrt[3]{2}}{8} + \frac{1}{16} + \frac{2 \sqrt[3]{9}}{81} = \frac{162 \sqrt[3]{2} + 81 + 32 \sqrt[3]{9}}{1296} 2) L . H . S = \frac{1}{(1 + \sqrt{2})} + \frac{1}{(\sqrt{2} + \sqrt{3})} + \frac{1}{(\sqrt{2} + \sqrt{3})} + \frac{1}{(\sqrt{2} + \sqrt{3})} + \frac{1}{(\sqrt{2} + \sqrt{3})} + \frac{1}{(\sqrt{2} + \sqrt{3})} + \frac{1}{(\sqrt{2} + \sqrt{3})} + \frac{1}{(\sqrt{2} + \sqrt{3})} = \frac{1}{(1 + \sqrt{2})} + 7 \times \frac{1}{(\sqrt{2} + \sqrt{3})} = \frac{1}{(1 + \sqrt{2})} \times \frac{1 - \sqrt{2}}{(1 - \sqrt{2})} + \frac{7}{(\sqrt{2} + \sqrt{3})} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} = \frac{1 - \sqrt{2}}{1 - 2} + \frac{7 (\sqrt{2} - \sqrt{3})}{2 - 3} = - (1 - \sqrt{2}) - (7 \sqrt{2} - 7 \sqrt{3}) = - 1 + \sqrt{2} - 7 \sqrt{2} + 7 \sqrt{3} = - 1 - 6 \sqrt{2} + 7 \sqrt{3} \neq 1 \neq R . H . S$

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Prove that :

$(i) (\sqrt{3 \times 5^{- 3}} \div \sqrt[3]{3^{- 1}} \sqrt{5}) \times \sqrt[6]{3 \times 5^{6}} = \frac{3}{5} (i i) 9^{\frac{3}{2}} - 3 \times 5^{0} - {(\frac{1}{81})}^{\frac{- 1}{2}} = 15 (i i i) {(\frac{1}{4})}^{- 2} - 3 \times 8^{\frac{2}{3}} \times 4^{0} + {(\frac{9}{16})}^{\frac{- 1}{2}} = \frac{16}{3} (i v) \frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{10^{\frac{- 1}{5}} \times 5^{\frac{3}{5}}} \div \frac{3^{\frac{4}{3}} \times 5^{\frac{- 7}{5}}}{4^{\frac{- 3}{5}} \times 6} = 10 (v) \sqrt{\frac{1}{4}} + (0.01)^{\frac{- 1}{2}} - (27)^{\frac{2}{3}} = \frac{3}{2} (v i) \frac{2^{n} + 2^{n - 1}}{2^{n + 1} - 2^{n}} = \frac{3}{2} (v i i) {(\frac{64}{125})}^{\frac{- 2}{3}} + \frac{1}{{(\frac{256}{625})}^{\frac{1}{4}}} + {(\frac{\sqrt{25}}{\sqrt[3]{64}})}^{0} = \frac{61}{16} (v i i i) \frac{3^{- 3} \times 6^{2} \times \sqrt{98}}{5^{2} \times \sqrt[3]{\frac{1}{25}} \times (15)^{\frac{- 4}{3}} \times 3^{\frac{1}{3}}} = 28 \sqrt{2} (i x) \frac{(0.6)^{0} - (0.1)^{- 1}}{{(\frac{3}{8})}^{- 1} {(\frac{3}{2})}^{3} + {(\frac{1}{3})}^{- 1}} = - \frac{3}{2}$

Q. Prove that

\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + . . . . . . . . . . . . . + \frac{1}{\sqrt{8} + \sqrt{9}}

Q. simplify:
i)

(1^{3} + 2^{3} + 3^{3})^{1 / 2}

ii)

{(\frac{81}{16})}^{3 / 4} \times {(\frac{25}{9})}^{- 3 / 2}

Q. Prove that

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

Q. Simplify:

\frac{1}{1 + \frac{\frac{2}{3}}{1 + \frac{2}{3} + \frac{\frac{8}{9}}{1 - \frac{2}{3}}}}

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Simplify: 1/(256)1/3 + 1/(256)1/2 + 2/(243)2/3Prove that : 1/(1+ √2) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3 )+ 1/(√2+ √3 )+ 1/(√2+ √3) = 2 .Prove that: {81/16}-3/2 x [ {25/9}-3/2 + {5/2}-1 = 1

Simplify: 1/(256)1/3 + 1/(256)1/2 + 2/(243)2/3
Prove that : 1/(1+ √2) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3) + 1/(√2+ √3 )
+ 1/(√2+ √3 )+ 1/(√2+ √3) = 2
.Prove that: {81/16}-3/2 x [ {25/9}-3/2 + {5/2}-1 = 1