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Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined.$\frac{-7+7 a}{21 a^{2}-21}$

$\frac{1}{3(a+1)}$$a \neq-1,1$

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 2

Simplifying Rational Expressions

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

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In mathematics, the absolu…

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All right. So we're giving this rational expression and were asked to put it in simplest form as well as list the value for this variable here that will make this expression undefined. So the first thing we're gonna do is take the denominator instead of equal to zero. So if I have 21 a squared minus 21 equal to zero as a I asked 21 to both sides, and then I divide by 21 and I get the A squared is equal to 21. Er sorry that a squared is equal to one. And then when I take the square root of both sides here, I get a is equal to. And remember, when I take the square root, I gotta have plus or minus one. Because if I have a negative one and plug this in here, you'll see negative one squared is gonna be a positive one. Which means that this will be 21 minus 21 which is zero, and then same with a positive one. It will yield the same result. So both a positive and a negative one will lead to an undefined expression. So basically a cannot be positive one or negative one for this expression, and then we're going to reduce it. So I just reorder the top up here. We almost always put the variables here first and in the numerator we see we can plot a seven and I haven't a minus one. And then I pull out a 21 and I'll have an A squared minus one. And so this seven in this 21 can be reduced to a one over three. And then here you see a perfect square minus with no middle term and another perfect square, So this could be factored. So I haven't a minus one in the numerator, and then I have a three. And remember, what this is is an A plus one times in a minus one. Because when I foiled this back out, when you have the exact same terms here with a plus and a minus, the middle term will cancel out. And then I'm left with this going backwards, right? So I multiply these out, I get this. So this is correct. And then you can see that those guys will cancel so that I'm left with the one in the numerator and then a three times in a plus one in the denominator

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