Question

How do you simplify `\frac { - 18p ^ { - 2} r ^ { 3} h ^ { - 8} v ^ { - 8} } { 9a ^ { - 1} p ^ { 3} r ^ { 3} v ^ { - 2} w }`?

Answer 1

`= -(2a)/(p^5h^8v^6w)`

Explanation:

`\frac { - 18p ^ { - 2} r ^ { 3} h ^ { - 8} v ^ { - 8} } { 9a ^ { - 1} p ^ { 3} r ^ { 3} v ^ { - 2} w }`

First, let's separate the like-terms to make it easier to see what's going on here:

`=(-18)/9 * 1/(a^(-1)) * p^-2/p^3 * r^3/r^3 * h^-8/1 * v^-8/v^-2 * 1/w`

`=(-18)/9 * a^0/(a^(-1)) * p^-2/p^3 * r^3/r^3 * h^-8/h^0 * v^-8/v^-2 * w^0/w`

since anything to the `0`th power is `1`.

The rules for like-terms and powers is:

• Multiplication of terms requires addition of powers
• Division of terms requires subtraction of powers

We will need to use the latter.

`=-2 * a^(0-(-1)) * p^(-2-3) * r^(3-3) * h^(-8-0) * v^(-8-(-2)) * w^(0-1)`

`=-2 * a^(1) * p^(-5) * r^(0) * h^(-8) * v^(-6) * w^(-1)`

`=-2ap^-5h^-8v^-6w^-1`

Rewriting with only positive powers (if you would like to):

`= -(2a)/(p^5h^8v^6w)`

Answer 2

`-(2a)/(p^5h^8v^6w)` or `-2ap^-5h^-8v^-6w^-1`

Explanation:

first, you can find individual quotients:

`18/9 = -2`

`1/(a^-1) = a^1 = a`

`p^-2/p^2 = p^(-2 - 3) = p^-5`

`r^3/r^3 = r^(3-3) = r^0 = 1`

`h^-8/1 = h^-8`

`v^-8/v^-2 = v^(-8 - -2) = v^(-8 + 2) = v^-6`

`1/w = w^-1`

multiplying all of these together gives a simplified version of the fraction.

hence, the expression can be written as `-2ap^-5h^-8v^-6w^-1`.

if you want to have the numerator as only positive exponents, you can write the negative exponents in the previous expression in the denominator.

this gives `-(2a)/(p^5h^8v^6w)`.