Question

How do you simplify `\frac { p ^ { - 1} q ^ { - 3} } { p ^ { 0} q ^ { 6} \cdot p ^ { - 9} q ^ { - 1} }`?

Answer 1

See the entire solution process below:

Explanation:

First, rewrite the expression as:

`(p^-1q^-3)/((p^0 * p^-9)(q^6 * q^-1))`

Next, use this rule for exponents on the two terms in the numerator:

`x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))`

`(p^-1q^-3)/((p^color(red)(0) * p^color(blue)(-9))(q^color(red)(6) xx q^color(blue)(-1))) = (p^-1q^-3)/(p^(color(red)(0) + color(blue)(-9))q^(color(red)(6) + color(blue)(-1))) = (p^-1q^-3)/(p^-9q^5)`

Then, rewrite the expression again as:

`(p^-1/p^-9)(q^-3/q^5)`

Now, use these rules of exponents to complete the simplification:

`x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))` and `x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))`

`(p^color(red)(-1)/p^color(blue)(-9))(q^color(red)(-3)/q^color(blue)(5)) = (p^(color(red)(-1)-color(blue)(-9)))(1/q^(color(blue)(5)-color(red)(-3))) = (p^(color(red)(-1)+color(blue)(9)))(1/q^(color(blue)(5)+color(red)(3))) =`

`p^8 * 1/q^8 = p^8/q^8`

Or

`(p/q)^8`