Question

How do you simplify `(x^-4y^-6z^-10 )/( a^1b^2c^-4)^2 * (a^1b c^-4) /( x^6y z^9)^2`?

Answer 1

`frac(c^4)(x^(16)*y^(8)*z^(28)*a^(1)*b^(3))`

Explanation:

As always, "count apples as apples and oranges as oranges".
In this case everything are really just multiplications and divisions.
If I see this, I would just open up the parentheses and see what can be removed. The remaining terms have to be the answer.

So let's attack this.
We have:
`frac(x^-4 y^-6 z^-10)((a^1 b^2 c^-4)^2)*frac(a^1 b c^-4)((x^6 y z^9)^2)`

Then, I would rewrite `b=b^1` (although it is unnecessary), and `y=y^1` just to be explicit.
Open the parentheses. Remember the rule: `(a^n)^m = a^(n*m)`
We get:
`frac(x^-4 y^-6 z^-10)(a^2 b^4 c^-8)*frac(a^1 b^1 c^-4)(x^12 y^2 z^18)`

Then it's easy, x goes with x, y goes with y, etc...
Remember the rule `frac(a^n)(a^m)=a^(n-m)`
So we have:
`x^(-4-12)*y^(-6-2)*z^(-10-18)*a^(1-2)*b^(1-4)*c^(-4+8)`
`=x^(-16)*y^(-8)*z^(-28)*a^(-1)*b^(-3)*c^4`
or rewriting it in fractional form (putting all the negative exponents at the bottom):

`=frac(c^4)(x^(16)*y^(8)*z^(28)*a^(1)*b^(3))`