Question

How do you simplify `\frac { p ^ { - 5} \cdot p ^ { - 3} } { p ^ { 4} \cdot p ^ { 0} }`?

Answer 1

See a solution process below:

Explanation:

First, use this rule of exponents to simplify the numerator and denominator:

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

`(p^color(red)(-5) * p^color(blue)(-3))/(p^color(red)(4) * p^color(blue)(0)) => p^(color(red)(-5)+color(blue)(-3))/p^(color(red)(4)+color(blue)(0)) => p^-8/p^4`

Now, use this rule of exponents to simplify the expression while ensuring all exponents are positive:

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

`p^color(red)(-8)/p^color(blue)(4) => 1/p^(color(blue)(4)-color(red)(-8)) => 1/p^(color(blue)(4)+color(red)(8)) => 1/p^12`

Answer 2

`1/p^12`

Explanation:

Note two of the laws of indices:

`color(blue)(x^0=1)" "(x!=0)" " and " "x^color(red)(-m) = 1/x^color(red)(m)`

So you can change any negative indices to positive indices.

`(color(red)(p^-5 xxp^-3))/(p^4xxcolor(blue)(p^0)) = 1/(color(red)(p^5xxp^3)xxp^4 xxcolor(blue)(1))`

Now add the indices of `p`

`=1/p^12`