Prove that-1-4-2-3 - 8 2-3- 40-9-16-1-2-16-3

( 1/4 )^ 2 3× 8^2/3× 4^0 +(9/16 )^ 1/2 1/4^ 2 3 (√(8))^2 × 4^∘ + ( 16/9 )^1/2 4^2 3(2)^2 +4/3 =48 36+4/3 =16/3 ...

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Standard IX

Mathematics

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Prove that:1/...

Question

Prove that:
${(\frac{1}{4})}^{- 2} - 3 \times 8^{\frac{2}{3}} \times 4^{0} + {(\frac{9}{16})}^{- \frac{1}{2}} = \frac{16}{3}$

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Solution

${(\frac{1}{4})}^{- 2} - 3 \times 8^{\frac{2}{3}} \times 4^{0} + {(\frac{9}{16})}^{- \frac{1}{2}} \frac{1}{4^{- 2}} - 3 (\sqrt[3]{8})^{2} \times 4^{\circ} + {(\frac{16}{9})}^{\frac{1}{2}} 4^{2} - 3 (2)^{2} + \frac{4}{3} = \frac{48 - 36 + 4}{3} = \frac{16}{3}$

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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{4}{3} m - \frac{3}{4} n)}^{2} + 2 m n = \frac{16}{9} m^{2} + \frac{9}{16} n^{2}

(\frac{1}{4})^{\frac{- 1}{2}} - 3 (8)^{\frac{2}{3}} + (\frac{9}{16})^{\frac{- 1}{2}}

Q. Prove that approximately, when x is very small,

\frac{3 {(x + \frac{4}{9})}^{\frac{1}{2}} {(1 - \frac{3}{4} x^{2})}^{\frac{1}{3}}}{2 {(1 + \frac{9}{16} x)}^{2}} = 1 - \frac{307}{256} x^{2}

Prove that:
(i) $\frac{1}{3 + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + 1} = 1$
(ii) $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} = 2$

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Prove that: (14)−2−3×823×40+(916)−12=163

(14)−2−3×823×40+(916)−1214−2−3(3√8)2×4∘+(169)1242−3(2)2+43=48−36+43=163

Prove that:
${(\frac{1}{4})}^{- 2} - 3 \times 8^{\frac{2}{3}} \times 4^{0} + {(\frac{9}{16})}^{- \frac{1}{2}} = \frac{16}{3}$

${(\frac{1}{4})}^{- 2} - 3 \times 8^{\frac{2}{3}} \times 4^{0} + {(\frac{9}{16})}^{- \frac{1}{2}} \frac{1}{4^{- 2}} - 3 (\sqrt[3]{8})^{2} \times 4^{\circ} + {(\frac{16}{9})}^{\frac{1}{2}} 4^{2} - 3 (2)^{2} + \frac{4}{3} = \frac{48 - 36 + 4}{3} = \frac{16}{3}$