3-3- 62-98-562 -1-25-15-4-3- 31-3

=3^ 3× 6^2 ×√(2 × 49)/5^2 × (1/25)^1/3× (3 × 5)^ 4/3× 3^1/3 =3^ 3× (2 × 3)^2 ×√(2)×√(49)/5^2 × (1/5^2)^1/3× 3^ 4/3× 5^ 4/3× 3^1/3 =3^ 3× 3^2 × 2^2 × 2^1/2× ...

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Byju's Answer

Standard XII

Mathematics

Properties of Cube Root of a Complex Number

3-3× 62×√98/5...

Question

$\frac{3^{- 3} \times 6^{2} \times \sqrt{98}}{562 \times \sqrt[3]{\frac{1}{25}} \times (15)^{- \frac{4}{3}} \times 3^{\frac{1}{3}}}$

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Solution

$= \frac{3^{- 3} \times 6^{2} \times \sqrt{2 \times 49}}{5^{2} \times (\frac{1}{25})^{\frac{1}{3}} \times (3 \times 5)^{- \frac{4}{3}} \times 3^{\frac{1}{3}}} = \frac{3^{- 3} \times (2 \times 3)^{2} \times \sqrt{2} \times \sqrt{49}}{5^{2} \times (\frac{1}{5^{2}})^{\frac{1}{3}} \times 3^{- \frac{4}{3}} \times 5^{- \frac{4}{3}} \times 3^{\frac{1}{3}}} = \frac{3^{- 3} \times 3^{2} \times 2^{2} \times 2^{\frac{1}{2}} \times 7}{5^{2} \times (5)^{- \frac{2}{3}} \times 5^{- \frac{4}{3}} \times 3^{- 4 \frac{4}{3}} \times 3^{\frac{1}{3}}} = \frac{3^{- 3 + 2} \times 2^{\frac{2 + 1}{2}} \times 7}{5^{2 - \frac{2}{5} . \frac{4}{3}} \times 3^{- \frac{4}{3} + \frac{1}{3}}}$
$= \frac{3^{- 1} \times 2^{\frac{5}{2}} \times 7}{5^{\circ} \times 3^{- 1}} = 2^{\frac{5}{2}} \times 7 = 2^{2 + \frac{1}{2}} \times 7 = 2^{2} \times 2^{\frac{1}{2}} \times 7 = 4 \times 7 \times 2^{\frac{1}{2}} = 28 \sqrt{2}$

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Prove that :

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