Question

How do you simplify `\frac { p q r ^ { 5} \cdot p ^ { - 1} q r ^ { 8} } { p ^ { 0} q r ^ { - 1} }`?

Answer 1

See a solution process below:

Explanation:

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

`a^color(red)(0) = 1`

`(pqr^5 * p^-1qr^8)/(p^color(red)(0)qr^-1) =>`

`(pqr^5 * p^-1qr^8)/(1 * qr^-1) =>`

`(pqr^5 * p^-1qr^8)/(qr^-1)`

Next, cancel the common term in the numerator and denominator:

`(pcolor(red)(cancel(color(black)(q)))r^5 * p^-1qr^8)/(color(red)(cancel(color(black)(q)))r^-1) =>`

`(pr^5 * p^-1qr^8)/r^-1`

Then, rewrite the numerator as:

`((p * p^-1)(r^5 * r^8)q)/r^-1`

Next, use these rules of exponents to simplify the numerator:

`a = a^color(red)(1)` and `x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))`

`((p^color(red)(1) * p^color(blue)(-1))(r^color(red)(5) * r^color(blue)(8))q)/r^-1 =>`

`(p^(color(red)(1)+color(blue)(-1))r^(color(red)(5)+color(blue)(8))q)/r^-1 =>`

`(p^color(red)(0)r^13q)/r^-1 =>`

`(1 * r^13q)/r^-1 =>`

`(r^13q)/r^-1`

Now, rewrite the expression and use this rule of exponents to complete the simplification:

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

`(r^color(red)(13)/r^color(blue)(-1))q =>`

`r^(color(red)(13)-color(blue)(-1))q =>`

`r^(color(red)(13)+color(blue)(1))q =>`

`r^14q` or `qr^14`