Question

How do you simplify `\frac { p ^ { - 4} q ^ { - 4} r ^ { 0} \cdot p ^ { - 1} r ^ { - 2} } { ( 2p ^ { 2} q ^ { - 2} r ^ { 0} ) ^ { 3} }`?

Answer 1

`q^2/(8*p^11*r^2)`

Explanation:

There are four rules you need to know. First, note that when you have 0 as your exponent, you are getting a result of 1 (assuming that number ≠ 0). This means:

`x^0=1, if x≠0`

The second is that when you raise an exponent to an exponent, you multiply the exponents. So:

`(x^a)^b=x^(a*b)`

When you multiply exponents, you add the exponents together. When you divide exponents, you subtract the bottom from the top. In other words:

`x^a*x^b=x^(a+b)`

`x^a/x^b=x^(a-b)`

If you are multiplying and dividing by exponents, you can combine both above to get:

`(x^a*x^b)/x^c=x^(a+b-c)`

Knowing these, we can simplify the expression you are given.

Since we have `r^0`, we can simplify that to 1. Let's expand the denominator as well.

`(p^-4*q^-4*r^0*p^-1*r^-2)/(2*p^2*q^-2*r^0)^3 ->`

`(p^-4*q^-4*1*p^-1*r^-2)/(2*p^2*q^-2*1)^3 ->`

`(p^-4*q^-4*p^-1*r^-2)/(2*p^2*q^-2)^3`

`(p^-4*q^-4*p^-1*r^-2)/(8*p^6*q^-6)`

Now that we got the first two rules out of the way, let's multiply and divide the exponents. Then we can simplify.

`(p^-4*q^-4*p^-1*r^-2)/(8*p^6*q^-6) ->`

`(p^(-4-1-6)*q^(-4+6)*r^-2)/8 ->`

`(p^-11*q^2*r^-2)/8 ->`

`q^2/(8*p^11*r^2)`