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Number Systems — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsNumber Systems2 Marks
Question
Simplify: $\frac{9^{\frac{1}{3}}\times27^{-\frac{1}2{}}}{3^\frac{1}{6}\times3^{-\frac{2}{3}}}$
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Answer

$\frac{9^{\frac{1}{3}}\times27^{-\frac{1}2{}}}{3^\frac{1}{6}\times3^{-\frac{2}{3}}}=\frac{{3^3}^{\frac{1}{3}}\times{3^3}^{-\frac{1}2{}}}{3^\frac{1}{6}\times3^{-\frac{2}{3}}}$ $[\because(\text{a}^\text{m})^\text{n}=\text{a}^\text{mn}]$ $=\frac{{3}^{\frac{2}{3}}\times{3}^{-\frac{3}{2}}}{3^\frac{1}{6}\times3^{-\frac{2}{3}}}=\frac{3^{\frac{2}{3}-\frac{3}{2}}}{3^{\frac{1}{6}-\frac{2}{3}}}$ $[\because\text{a}^\text{m}\times\text{a}^\text{n}=\text{a}^\text{m+n}]$ $\frac{3^{\frac{4-9}{6}}}{3^{\frac{1-4}{6}}}=\frac{3^{-\frac{5}{6}}}{3^{-\frac{3}{6}}}=3^{-\frac{5}{6}+\frac{3}{6}}=3^{-\frac{2}{6}}=3^{-\frac{1}{3}}$ $\Big[\because\frac{\text{a}^\text{m}}{\text{a}^\text{n}}=\text{a}^{\text{m}-\text{n}}\Big]$

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