Question

How do you simplify `\frac { - 9a ^ { 9} b c ^ { 2} d ^ { 7} } { - 6a c ^ { 12} d ^ { 7} }`?

Answer 1

See a solution process below:

Explanation:

First, rewrite the expression as:

`(-9)/-6(a^9/a)b(c^2/c^12)(d^7/d^7) =>`

`(-3 xx 3)/(-3 xx 2)(a^9/a)b(c^2/c^12)(color(red)(cancel(color(black)(d^7)))/color(red)(cancel(color(black)(d^7)))) =>`

`(color(red)(cancel(color(black)(-3))) xx 3)/(color(red)(cancel(color(black)(-3))) xx 2)(a^9/a)b(c^2/c^12)1 =>`

`3/2(a^9/a)b(c^2/c^12)`

Next, use these rules of exponents to simplify the `a` terms:

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

`3/2(a^color(red)(9)/a^color(blue)(1))b(c^2/c^12) =>`

`3/2a^(color(red)(9)-color(blue)(1))b(c^2/c^12) =>`

`3/2a^8b(c^2/c^12) =>`

`(3a^8b)/2(c^2/c^12)`

Now, use this rule of exponents to simplify the `c` term:

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

`(3a^8b)/2(c^color(red)(2)/c^color(blue)(12)) =>`

`(3a^8b)/2(1/c^(color(blue)(12)-color(red)(2))) =>`

`(3a^8b)/2(1/c^10) =>`

`(3a^8b)/(2c^10)`