Simplify each complex fraction.

Question

Simplify each complex fraction. $\frac{\frac{6}{x^2 - 25} + x}{\frac{1}{x - 5}}$

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we want to simplify the expression seven divided by p squared minus 100 plus P Divided by one divided by P -10. So in this case you can see that this expression is really a big fraction where we have fractions in the numerator but also a fraction in the denominator. In order to make it easier to simplify. I'm going to first just take a look at the numerator. So the numerator seven divided by p squared minus 100 plus P. Well in this case we have this fraction and then this variable standing alone. So we want to try to combine these two components and we can do that by making P the second component have a denominator of p squared minus 100. Which we can do by multiplying P with p squared minus 100 divided by p squared minus 100. And you can see that P squared minus 100 over P squared minus evaluates to one. So that makes sure that we don't introduce any new numbers into our expression. But if we do this multiplication we now have seven divided by p squared minus 100. That first component didn't change plus p times p squared minus 100 divided by p squared minus 100. And now you can see that they have a common denominator so we can combine the two fractions if I do that I have seven plus I distribute the p p times p squared is p two. The third minus p times is 100 P. And that's being divided by the common denominator of p squared minus 100. And if we look at p squared minus 100 you can see that. I can write this as p squared minus squared since 10 squared is the same as 100. And when you see it in this form you can see that it resembles a difference of squares. Where if I had Y squared minus X squared, I can write this as Y plus X times y minus X. So if I do that same knowledge two p squared minus 10 squared, I can write this as P plus Times P -10. So my numerator will be P to the third minus P plus seven. You can see I rearranged the numerator with my denominator being p plus 10 times P minus 10. And you may be wondering why I decided to rewrite it, rewrite the denominator in that format. What's because we still have to deal with the denominator of the big fraction. So if I recombine my numerator into the big fraction, you can see that this becomes P to the 3rd -100 p plus seven, divided by P plus 10 times p minus 10 divided by That denominator which is one divided by P -10. Well, because I still have this big fraction with fractions in the numerator and denominator, I'm going to write this in the linear form for division, where I'll be dividing by One divided by P -10. So you see I just move the fraction in the denominator up to this line. And in this linear format I can change this division to be multiplication. If I switch the numerator and denominator of the second fraction, so if I switch the numerator and denominator of my second fraction, it becomes P -10 divided by one. And from here you can see that I have P -10 in the denominator of my first fraction. So that I can actually cancel out P -10. And that leaves me with P to the third minus 100. P plus seven divided by P plus 10. And you can see I can't simplify this any further. So this becomes my final simplification for the expression. And if we look at the answer choice choices, answer choice B also has P to the third minus 100 P plus seven divided by P plus 10. So, flee this video helped and see you next time

Simplify each complex fraction. $\frac{\frac{6}{x^2 - 25} + x}{\frac{1}{x - 5}}$

Key Concepts

Complex Fractions

Factoring Quadratic Expressions

Operations with Rational Expressions

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Simplify each complex fraction. 6x2−25+x1x−5\frac{\frac{6}{x^2 - 25} + x}{\frac{1}{x - 5}}x−51x2−256+x

Key Concepts

Complex Fractions

Factoring Quadratic Expressions

Operations with Rational Expressions

Watch next

Simplify each complex fraction. $\frac{\frac{6}{x^2 - 25} + x}{\frac{1}{x - 5}}$