Answer the question in attachment

1.	Answer the question in attachment
Answer» $\bf{\large{\underline{\underline{Question:-}}}}$ $\large{ \sf Simplify \: \dfrac{ {9}^{\frac{1}{3}} \times {27}^{ - \frac{1}{2} } }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ $\bf{\large{\underline{\underline{Solution:-}}}}$ $\large{\sf \dfrac{ {9}^{\frac{1}{3}} \times {27}^{ - \frac{1}{2} } }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ $\large{ \sf = \dfrac{ {({3}^{2})}^{\frac{1}{3}} \times {( {3}^{3})}^{ - \frac{1}{2} } }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ [In NUMERATOR $\sf {9}^{\frac{1}{3}} = {({3}^{2})}^{\frac{1}{3}}$ because 9 can be written as 3² and $\sf {27}^{-\frac{1}{2}} = {({3}^{3})}^{-\frac{1}{2}}$ because 27 can be written as 3³] $\large{ \sf = \dfrac{ {(3)}^{\frac{1}{3} \times 2} \times {(3)}^{ - \frac{1}{2} \times3 } }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ [LAW of exponents used in numerator $\sf (a^m)^n = a^{mn}$ ] $\large{ \sf = \dfrac{ {(3)}^{\frac{2}{3}} \times {(3)}^{ - \frac{3}{2}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ $\large{ \sf = \dfrac{ {(3)}^{\frac{2}{3}+ ( - \frac{3}{2})} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ [Law of exponents used in numerator $\sf a^m \times a^n = a^{m + n}$ ] $\large{ \sf = \dfrac{ {(3)}^{\frac{2}{3}- \frac{3}{2}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ $\large{ \sf = \dfrac{ {(3)}^{\frac{2 \times 2}{3 \times 2}- \frac{3 \times 3}{2 \times 3}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ $\large{ \sf = \dfrac{ {(3)}^{\frac{4}{6}- \frac{9}{6}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ $\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$ $\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{1}{8} + ( - \frac{2}{3})}} }$ [Law of exponents used in numerator $\sf a^m \times a^n = a^{m + n}$ ] $\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{1}{8}- \frac{2}{3}}} }$ $\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{1 \times 3}{8 \times 3}- \frac{2 \times 8}{3 \times 8}}} }$ $\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{3}{24}- \frac{16}{24}}} }$ $\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{- \frac{13}{24}}} }$ $\large{ \sf =(3)^{ - \frac{5}{6} - ( - \frac{13}{24}) } }$ [Law of EXPONENT used $\sf a^m \div a^n = a^{m-n}$ ] $\large{ \sf =(3)^{ - \frac{5}{6} + \frac{13}{24}} }$ $\large{ \sf =(3)^{ - \frac{5 \times 4}{6 \times 4} + \frac{13}{24}} }$ $\large{ \sf =(3)^{ - \frac{20}{24} + \frac{13}{24}} }$ $\large{ \sf =3^{ - \frac{7}{24}} }$ $\Huge{ \sf = \dfrac{1}{ {3}^{ \frac{7}{24} } } }$ [Law of exponent used $\sf a^{-m} = \dfrac{1}{a^m}$ ]

Answer»

$\bf{\large{\underline{\underline{Question:-}}}}$

$\large{ \sf Simplify \: \dfrac{ {9}^{\frac{1}{3}} \times {27}^{ - \frac{1}{2} } }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

$\bf{\large{\underline{\underline{Solution:-}}}}$

$\large{\sf \dfrac{ {9}^{\frac{1}{3}} \times {27}^{ - \frac{1}{2} } }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

$\large{ \sf = \dfrac{ {({3}^{2})}^{\frac{1}{3}} \times {( {3}^{3})}^{ - \frac{1}{2} } }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

[In NUMERATOR $\sf {9}^{\frac{1}{3}} = {({3}^{2})}^{\frac{1}{3}}$ because 9 can be written as 3² and $\sf {27}^{-\frac{1}{2}} = {({3}^{3})}^{-\frac{1}{2}}$ because 27 can be written as 3³]

$\large{ \sf = \dfrac{ {(3)}^{\frac{1}{3} \times 2} \times {(3)}^{ - \frac{1}{2} \times3 } }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

[LAW of exponents used in numerator $\sf (a^m)^n = a^{mn}$ ]

$\large{ \sf = \dfrac{ {(3)}^{\frac{2}{3}} \times {(3)}^{ - \frac{3}{2}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

$\large{ \sf = \dfrac{ {(3)}^{\frac{2}{3}+ ( - \frac{3}{2})} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

[Law of exponents used in numerator $\sf a^m \times a^n = a^{m + n}$ ]

$\large{ \sf = \dfrac{ {(3)}^{\frac{2}{3}- \frac{3}{2}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

$\large{ \sf = \dfrac{ {(3)}^{\frac{2 \times 2}{3 \times 2}- \frac{3 \times 3}{2 \times 3}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

$\large{ \sf = \dfrac{ {(3)}^{\frac{4}{6}- \frac{9}{6}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

$\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{1}{8}} \times {3}^{ - \frac{2}{3} } } }$

$\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{1}{8} + ( - \frac{2}{3})}} }$

[Law of exponents used in numerator $\sf a^m \times a^n = a^{m + n}$ ]

$\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{1}{8}- \frac{2}{3}}} }$

$\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{1 \times 3}{8 \times 3}- \frac{2 \times 8}{3 \times 8}}} }$

$\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{ \frac{3}{24}- \frac{16}{24}}} }$

$\large{ \sf = \dfrac{ {(3)}^{- \frac{5}{6}} }{ {3}^{- \frac{13}{24}}} }$

$\large{ \sf =(3)^{ - \frac{5}{6} - ( - \frac{13}{24}) } }$

[Law of EXPONENT used $\sf a^m \div a^n = a^{m-n}$ ]

$\large{ \sf =(3)^{ - \frac{5}{6} + \frac{13}{24}} }$

$\large{ \sf =(3)^{ - \frac{5 \times 4}{6 \times 4} + \frac{13}{24}} }$

$\large{ \sf =(3)^{ - \frac{20}{24} + \frac{13}{24}} }$

$\large{ \sf =3^{ - \frac{7}{24}} }$

$\Huge{ \sf = \dfrac{1}{ {3}^{ \frac{7}{24} } } }$

[Law of exponent used $\sf a^{-m} = \dfrac{1}{a^m}$ ]

Answer the question in attachment

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