Question 1092034
<br>There are many different paths to the final simplified form.  Different students will have preferences for different steps to do first....<br>
The given expression is<br>
{{{((((3)(x^3)(y^2))/((2)(z^5)))^3)*((((-4)(z^8))/((9)(x^4)(y^3)))^3)}}}<br>
I put the coefficients 3, 2, -4, and 9 in parentheses to emphasize that they, as well as the variables and their exponents, need to be raised to the prescribed powers.  Many beginning students fail to realize that.<br>
Since there are two complex fractions being mutliplied together, we can combine all the factors in the numerators and denominators of the two fractions, raised to the appropriate powers, in a single fraction.<br>
{{{((3^3)(x^9)(y^6)((-4)^3)(z^24))/((2^3)(z^15)(9^3)(x^12)(y^9))}}}
For the numerical part the simplified form, we have
{{{(3^3)/(9^3) = 1/3^3 = 1/27}}}  and
{{{((-4)^3)/(2^3) = (-2)^3 = -8}}}
so the numerical part of the simplified form is -8/27.<br>
Then simplifying the exponents on the variables is easy, using basic rules.<br>
x: 9 factors in the numerator, 12 in the denominator --> 3 factors left in the denominator<br>
y: 6 factors in the numerator, 9 in the denominator --> 3 factors left in the denominator<br>
z: 24 factors in the numerator, 15 in the denominator --> 9 factors left in the numerator<br>
And the final simplified form is
{{{(-8z^9)/(27(x^3)(y^3))}}}