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

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)#

#{ (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)#