Simplify each complex fraction.

Question

Simplify each complex fraction. $\frac{m - \frac{1}{m^2 - 4}}{\frac{1}{m + 2}}$

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we want to simplify the expression that is a really big fraction with N plus one divided by n squared minus 81 in the numerator and one divided by n minus nine in the denominator. So in this expression you can see that we have fractions in both the numerator and the denominator. So I'm gonna go ahead and solve them separately or simplify them separately and then combine them to make it easier to simplify. So I'll start with my numerator which is N plus one divided by N squared minus 81. And in this case you can see that this second component is a fraction and this first component is just the variable end. So in order to combine the two, I have to introduce a common denominator into my into both components. So because N has a denominator of one right now I can introduce and squared - into. And by multiplying it with and squared minus 81 over N squared minus 81. And recall that I need to multiply and squared minus 81 over itself because this simplifies to one. So I'm technically multiplying N times one. So this ensures that I don't introduce any new numbers into my expression. So when I do this multiplication, I have n squared I have n times n squared which is N to the third, N times negative 81 is negative 81. N Divided by N Squared -81. And then the second fraction stays the same with one divided by n squared minus 81. And you can see that as a result of the multiplication. We now have the same denominator in both of these components. So I can go ahead and combine them. So when I combine them it becomes N. To the third minus 81 N plus one, divided by n squared minus 81. The common denominator. And I can go ahead and reintroduce my denominator of one divided by n minus n and minus nine. So because we still have fractions and the numerator and denominator, I'm going to go ahead and write this in the more linear format or division. Where my denominator will be the second fraction. And in this linear form, a recall that I can change this division to multiplication. As long as I switch the numerator and denominator of the second fraction. So that means that one divided by N -9 will become N -9 divided by one. And from here it may seem like you can't simplify this any further, but when we think of 81 can think of it as nine times nine or nine squared. So that means that I can write n squared minus as n squared minus nine squared and n squared minus nine squared looks very similar to the format for difference of squares, where we can write x squared minus Y squared sex plus Y, times x minus Y. So I apply that same format to n squared minus a squared and squared minus nine squared. I can write it as an plus nine times and minus nine. So that means I can plug this into my denominator and once I plugged that into my or if I replace the and squared minus 81 with N plus nine times n minus nine. You can see that I have a n minus nine in the denominator of my first fraction and the numerator of my second fraction, which means that I can cancel them out. So that leaves me with and to the third -81, n plus one divided by N. Plus nine. And I can't simplify this any further. So this becomes my final simplification. And if we look at the answer choices, you can see that answer choice A. Also has N. To the third minus 81. N plus one divided by N plus nine. So hopefully this video helped and see you next time.

Simplify each complex fraction. $\(\frac{m - \frac{1}{m^2 - 4}\)}{\(\frac{1}{m + 2}\)}$

Key Concepts

Complex Fractions

Factoring Quadratic Expressions

Multiplying by the Reciprocal

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