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Why does the denominator change to 9

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I'm not sure where the 27^x⅔ comes from to be honest...

27^2/3 = 9

That's where your 9 denominator comes from...
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Because taking the 2/3rd power of something is equivalent to taking the cube root, then squaring (or squaring, then taking the cube root). So

27^2/3 = (27^(1/3))^2 = 3^2 = 9 or    
27^2/3 = (27^(2))^1/3 = 729^1/3 = 9

Another way to see it is to write 27 as 3^(3). Then you can apply the same logic that transformed x^27 to x^(18).
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The exponent, when outside of parentheses, applies to both top and bottom. 27^(2/3) = (27^(1/3))^2 = 3^2 = 9
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27 = 3\^3

(27)\^2/3 = (3\^3)\^(2/3) = 3\^(6/3) = 3\^2 = 9

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