Question

Simplify:

$fraction numerator 9 to the power of begin display style 1 third end style end exponent cross times 27 to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent cross times 3 to the power of begin display style 1 third end style end exponent end fraction space space space$

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Solution

fraction numerator 9 to the power of begin display style 1 third end style end exponent cross times 27 to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent cross times 3 to the power of begin display style 1 third end style end exponent end fraction space space space space space equals space fraction numerator left parenthesis 3 squared right parenthesis to the power of begin display style 1 third end style end exponent cross times left parenthesis 3 cubed right parenthesis to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 plus 1 third end style end exponent end fraction space space space space space equals space fraction numerator 3 to the power of 2 cross times begin display style 1 third end style end exponent straight x 3 to the power of 3 straight x begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent end fraction space space space space equals space 3 to the power of begin display style 2 over 3 plus 3 over 2 end style end exponent over 3 to the power of begin display style 1 over 6 end style end exponent space space space space equals space space 3 to the power of begin display style 13 over 6 end style end exponent over 3 to the power of begin display style 1 over 6 end style end exponent space space space space equals space 3 squared space space space space equals space 9

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