Simplify:frac{9^{frac{1}{3}} times 27^{-frac{ | vedclass

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Question

(D) To simplify the expression $\frac{9^{\frac{1}{3}} 	imes 27^{-\frac{1}{2}}}{3^{\frac{1}{6}} 	imes 3^{-\frac{2}{3}}}$,we first express all bases a

Vedclass · Accepted Answer

$3^{-\frac{1}{3}}$

Vedclass · Answer

(D) To simplify the expression $\frac{9^{\frac{1}{3}} 	imes 27^{-\frac{1}{2}}}{3^{\frac{1}{6}} 	imes 3^{-\frac{2}{3}}}$,we first express all bases as powers of $3$.Step $1$: Rewrite the bases:$9 = 3^2$ and $27 = 3^3$.Step $2$: Substitute these into the expression:$\frac{(3^2)^{\frac{1}{3}} 	imes (3^3)^{-\frac{1}{2}}}{3^{\frac{1}{6}} 	imes 3^{-\frac{2}{3}}}$Step $3$: Apply the power rule $(a^m)^n = a^{m 	imes n}$:$\frac{3^{\frac{2}{3}} 	imes 3^{-\frac{3}{2}}}{3^{\frac{1}{6}} 	imes 3^{-\frac{2}{3}}}$Step $4$: Apply the product rule $a^m 	imes a^n = a^{m+n}$ for the numerator and denominator:Numerator: $3^{\frac{2}{3} - \frac{3}{2}} = 3^{\frac{4-9}{6}} = 3^{-\frac{5}{6}}$Denominator: $3^{\frac{1}{6} - \frac{2}{3}} = 3^{\frac{1-4}{6}} = 3^{-\frac{3}{6}} = 3^{-\frac{1}{2}}$Step $5$: Apply the quotient rule $\frac{a^m}{a^n} = a^{m-n}$:$3^{-\frac{5}{6} - (-\frac{1}{2})} = 3^{-\frac{5}{6} + \frac{3}{6}} = 3^{-\frac{2}{6}} = 3^{-\frac{1}{3}}$Thus,the simplified form is $3^{-\frac{1}{3}}$.

Simplify:
$\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{\frac{1}{6}} \times 3^{-\frac{2}{3}}}$

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Simplify:$\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{\frac{1}{6}} \times 3^{-\frac{2}{3}}}$