Simplify each complex fraction. [ y + 1/(y^2-9) ] / [ 1/(y + 3) ]

Question

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we want to simplify this expression that is a big fraction. Where we have B minus one divided by B squared minus 144 in the numerator and one divided by B -12 in the denominator. So in this case you can see that we have a complex expression with a fraction that has fractions in the numerator and in the denominator. So in order to make it easier to simplify, I'm going to deal with the numerator and denominator separately and then combine them. So if I look at the numerator, I have B minus one divided by B squared minus 144. So I have this fraction and this be by itself. So in order to simplify, I want to try and combine these two components and I can do that by making sure that this first component, B has a denominator of B squared minus 140 for which I can introduce by multiplying B with B squared minus 144 Divided by B. -144. So notice that I have to multiply by B squared minus 140 for over itself because this simplifies to one. So if I multiply one times B, I'm making sure that I'm not introducing any new numbers into my expression. So once I do this multiplication, I have B squared times B is B to the third negative 144 times B is negative 144 B divided by B squared - minus one divided by B squared minus 144. And after the multiplication you can see that we now have the same denominator which means I can combine these two components together. So if I combine the two components together, I now have B to the third minus B minus one divided by that common denominator of B squared minus 144. So from here I can reintroduce my denominator of One divided by B -12. So because I still have fractions in the numerator and the denominator, I'm going to write this division in the linear format. So I have this be This one divided by B minus 12 B. The second fraction. And in this linear format recall that I can change this division to be multiplication. As long as I switch the numerator and denominator of this second fraction. So instead of being one divided by B -12, I have B -12 divided by one. And from here it may seem like you can't have any more simplification, But if you think of 144, this is the same as 12 times 12 or squared, which means I can write B squared minus as b squared minus 12 squared. And this follows the form of the difference of squares very closely. So when you think of difference of squares, you have x squared minus Y squared and I can expand this to be X plus Y, times x minus y. So if I apply that same knowledge to B squared minus 12 squared, I can write it as B plus 12 times B minus 12. So if I substitute this expression in for B squared minus you can now see that I have a B - and the denominator of this first fraction and the numerator of the second fraction. So I can cancel them out, Leaving me with just be to the 3rd -144, B -1 Divided by B Plus 12. And now I can't simplify this any further. So this becomes my final simplification for the expression. And if I look at the answer choices, you can see that B also has B to the third minus 144 B minus one divided by B plus 12. So hopefully this video helped and see you next time

Simplify each complex fraction. [ y + 1/(y^2-9) ] / [ 1/(y + 3) ]

Key Concepts

Complex Fractions

Factoring Polynomials

Reciprocal of a Fraction

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