Question

How do you simplify `\frac { 24a ^ { - 3} b ^ { 4} c ^ { - 7} } { 6a ^ { 6} b ^ { - 1} }`?

Answer 1

See the entire solution process below:

Explanation:

First, rewrite this expression as:

`(24/6)(a^-3/a^6)(b^4/b^-1)c^-7 = 4(a^-3/a^6)(b^4/b^-1)c^-7`

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

`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))`

`4(a^color(red)(-3)/a^color(blue)(6))(b^color(red)(4)/b^color(blue)(-1))c^-7 = 4(1/a^(color(blue)(6)-color(red)(-3)))(b^(color(red)(4)-color(blue)(-1)))c^-7 =`

`4(1/a^(color(blue)(6)+color(red)(3)))(b^(color(red)(4)+color(blue)(1)))c^-7 = 4(1/a^9)(b^5)c^-7`

Now, use this rule of exponents to eliminate the negative exponent on the `c` term:

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

`4(1/a^9)(b^5)c^color(red)(-7) = 4(1/a^9)(b^5)1/c^color(red)(- -7) = 4(1/a^9)(b^5)1/c^color(red)(7)`

We can now multiply the individual terms:

`(4b^5)/(a^9c^7)`