Question

How do you combine `\frac { 24w z } { w ^ { 2} - 36z ^ { 2} } + \frac { w - 6z } { w + 6z }` into one fraction?

Answer 1

`(w+6z)/(w-6z)`

Explanation:

Notice the denominator of the expression on the left hand side can be factored as

`w^2-36z^2=(w-6z)(w+6z)`

Then we can rewrite the original question as

`=(24wz)/((w+6z)(w-6z))+(w-6z)/(w+6z)`

To combine fractions, the denominators must be the same. So multiplying the numerator and denominator of the right hand expression by the term `(w-6z)` gives

`=(24wz)/((w+6z)(w-6z))+(w-6z)/(w+6z)(w-6z)/(w-6z)`

Now, with the denominators the same, we can add the fractions

`=(24wz+(w-6z)(w-6z))/((w-6z)(w+6z))`

Expanding the numerator gives

`=(24wz+w^2-12wz+36z^2)/((w-6z)(w+6z))`

Combine like terms

`=(12wz+w^2+36z^2)/((w-6z)(w+6z))`

Rewrite so you can see the numerator factors into a perfect square

`=(w^2+12wz+36z^2)/((w-6z)(w+6z))=((w+6z)cancel((w+6z)))/((w-6z)cancel((w+6z)))`

ANSWER: `(w+6z)/(w-6z)`

Answer 2

multiply the second terms by ( x - 6z) so both terms have the same denominator so that they can be added.

Explanation:

`{ (w-6z) xx (w-6z)}/{(w +6z)xx (w-6z)} = (w^2 -12 wz + 36z^2)/(w^2-36z^2)`

The second term now has the same denominator as the first term so they can be added.

`24wz + (w^2 -12z + 36z^2)/ (w^2 - 36z^2) = (w^2 + 12wz + 36z^2)/(w^2-36z^2)`

The numerator and denominator can be factored and common terms divided out.

`(w^2 + 12 wz + 36z)/ (w^2 - 36z)`= `{ (w + 6z)(w+6z)}/{(w+6z)(w-6z)}`

the common term # (w + 6z) can be divided out leaving

`( w + 6z) / (w -6z)`