Question

How do you simplify `(27a^-3b^12)^(1/3) / (16a^-8b^12) ^ (1/2)`?

Answer 1

You must remember the exponent-radical rule `x^(1/n) = root(n)(x)`

Explanation:

Therefore,

`(root(3)(27) xx (a^-3 xx b^12)^(1/3))/(sqrt(16) xx (a^-8 xx b^12)^(1/2))`

For the expressions in parentheses, you must calculate using the power of exponents rule, or `(a^n)^m = a^(n xx m)`

`=(3 xx a^-1 xx b^4)/(4 xx a^-4 xx b^6)`

However, we need to simplify further and write without negative exponents. This can all be done using the quotient rule `a^n/a^m = a^(n- m)`

As a shortcut to not have to use the negative exponent rule `a^-n = 1/a^n`, we must apply the quotient rule from the largest exponent. For example, in `x^2/x^4`, you would make `x^4` as n and `x^2` as m, and then you would do your subtraction. You would get `1/x^2` in this problem, which is without negative exponents, and is what we want.

`= (3 xx a^(-1 - (-4)) )/(4 xx b^(6 -4))`

`= (3a^3)/(4b^2)`

Hopefully this helps!