Question

Answer the question in attachment

Answer 1

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

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

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

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

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

[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 2

Answer:

$\dfrac{9^{\dfrac{1}{3}}\times \:27^{-\dfrac{1}{2}}}{3^{\dfrac{1}{8}}\times \:3^{-\dfrac{2}{3}}}=\dfrac{1}{3^{\dfrac{7}{24}}}\quad \left(\mathrm{Decimal:\quad }\:0.72584\dots \right)$

Step-by-step explanation:

$\dfrac{9^{\dfrac{1}{3}}\times \:27^{-\dfrac{1}{2}}}{3^{\dfrac{1}{8}}\times \:3^{-\dfrac{2}{3}}}$

$\gray{3^{\dfrac{1}{8}}\times \:3^{-\dfrac{2}{3}}=3^{\dfrac{1}{8}-\dfrac{2}{3}}}$

$=\dfrac{27^{-\dfrac{1}{2}}\times \:9^{\dfrac{1}{3}}}{3^{\dfrac{1}{8}-\dfrac{2}{3}}}$

$\gray{\mathrm{Factor}\:9^{\dfrac{1}{3}}:\quad 3^{\dfrac{2}{3}}}$

$=\dfrac{27^{-\dfrac{1}{2}}\times \:3^{\dfrac{2}{3}}}{3^{\dfrac{1}{8}-\dfrac{2}{3}}}$

$\gray{\mathrm{Cancel\:}\dfrac{3^{\dfrac{2}{3}}\times \:27^{-\dfrac{1}{2}}}{3^{\dfrac{1}{8}-\dfrac{2}{3}}}}$

$\dfrac{3^{\dfrac{2}{3}}\times \:27^{-\dfrac{1}{2}}}{3^{\dfrac{1}{8}-\dfrac{2}{3}}}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \dfrac{x^a}{x^b}=x^{a-b}}$

$\gray{\dfrac{3^{\dfrac{2}{3}}}{3^{\dfrac{1}{8}-\dfrac{2}{3}}}=3^{\dfrac{2}{3}-\left(\dfrac{1}{8}-\dfrac{2}{3}\right)}}$

$=27^{-\dfrac{1}{2}}\times \:3^{\dfrac{2}{3}-\left(\dfrac{1}{8}-\dfrac{2}{3}\right)}$

$\gray{\mathrm{Subtract\:the\:numbers:}\:\dfrac{2}{3}-\left(\dfrac{1}{8}-\dfrac{2}{3}\right)=\dfrac{29}{24}}$

$=3^{\dfrac{29}{24}}\times \:27^{-\dfrac{1}{2}}$

$\gray{\mathrm{Factor\:integer\:}27=3^3]$

$=3^{\dfrac{29}{24}}\left(3^3\right)^{-\dfrac{1}{2}}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}}$

$\gray{\left(3^3\right)^{-\dfrac{1}{2}}=3^{3\left(-\dfrac{1}{2}\right)}}$

$=3^{\dfrac{29}{24}}\times \:3^{3\left(-\dfrac{1}{2}\right)}$

$\gray{\mathrm{Refine}}$

$=3^{\dfrac{29}{24}}\times \:3^{-\dfrac{3}{2}}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \:a^b\times \:a^c=a^{b+c}}$

$\gray{3^{\dfrac{29}{24}}\times \:3^{-\dfrac{3}{2}}=\:3^{\dfrac{29}{24}-\dfrac{3}{2}}}$

$=3^{\dfrac{29}{24}-\dfrac{3}{2}}$

$\gray{\mathrm{Join}\:\dfrac{29}{24}-\dfrac{3}{2}:\quad -\dfrac{7}{24}}$

$=3^{-\dfrac{7}{24}}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \:a^{-b}=\dfrac{1}{a^b}}$

$=\dfrac{1}{3^{\dfrac{7}{24}}}$