Simplify the rational expressions below: 6 x 12 x \frac{6x}{12x}

In recent videos, we got introduced to rational expressions. We talked about what these are and how they are defined. Now in this video, we're going to learn about how we can simplify rational expressions. And I think you're gonna find this process is actually relatively straightforward to what we already know about simplifying rational numbers. So let's just go ahead and get right into things. Now, first off, I want you to recall how we could simplify a rational number, which is also just simplifying a fraction. What you first wanna do is take your fraction and factor it completely. So factoring the top 28, this will give us seven times four. And factoring the bottom 35, we get seven times five. Notice how seven seven is a common factor, which we can cancel, giving us the simplified answer of four over five. Now, it turns out simplifying rational expressions is actually very similar. What you want to do is factor the numerator and denominator completely, and from there, you're going to cancel the common factors. So let's go ahead and try simplifying this rational expression. Now 28, I know that's the same thing as seven times four. We actually factored 28 over here, but seven times four is broken down into prime numbers. And then for this x cubed, well, that's just the same thing as x times x times x. So, this is what we get on top. Now, on bottom, in the denominator, we have 35. 35 breaks down into seven times five, and then notice we have x to the fifth, which is the same thing as x times x times x times x times x. So, now we have this fully factored. Now, from here, what you want to do is cancel the common factors. Now, I can see we have a seven in common in the numerator and denominator. So what I can do is cancel the sevens right here. Now, notice also I have three x's in the numerator and three x's in the denominator, which are going to cancel each other completely. So what that means is in the numerator, we'll have four left over, and then in the denominator, we have a five left over, as well as two more x's, which I can also just write as x squared. So four over five x squared is the simplified answer to this rational expression. So this is how you can simplify rational expressions. Now I wanna make sure we can do this in an example as well, so let's try a couple more examples here to really make sure we have this down. Now this problem asks us to simplify the following expressions, and we'll start with example a. Notice here that everything is already factored. We have x plus two times x minus two on top, with an x plus six times x plus two on bottom. Since this is already factored, we can just cancel the common factors, which I can see would be x plus two. So that means in the numerator, we'll only have x minus two left over, and then in the denominator, we'll only have x plus six left over. And this would be our simplified answer, and that's it. So as you can see, when things are already factored, it's very straightforward. But now let's take a look at example b. In this example, notice that things are not fully factored here. I mean, we have the x minus five, which is fully factored. We have a three x squared minus 15 x, which is not. But that's okay because we can factor this completely. We'll keep the x minus five on top, and then on the bottom, I'm going to pull out the greatest common factor, which is three x. If I pull a three x to the outside here, inside parentheses, we'll have three x times whatever would give us x squared, which I know is just x, and that's going to be minus, and that's going to be three x times whatever would give us 15 x, which is five. Notice if I redistributed this three x into the parentheses, I would get the same thing that we started with. So we did this correctly. Now the reason that I factored it like so is notice we now have the common factor x minus five, which is going to cancel. So all we're gonna end up getting here is one over three x, and that right there is the simplified answer and the solution to this problem. So this is how you can simplify rational expressions. Hope you found this video helpful, and definitely check out the practice and examples after this video to make sure we have this concept down. See you all in the next one.

8. Rational Expressions and Functions

Simplifying Rational Expressions

Intermediate Algebra

8. Rational Expressions and Functions

Simplifying Rational Expressions

Struggling with Intermediate Algebra?

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Multiple Choice

Simplify the rational expressions below:
$6 x over 12 x$

$2x$

$2$

$\frac12x$

$\frac12$

Verified step by step guidance

Start with the given rational expression: $\frac{6x}{12x}$.

Identify common factors in the numerator and denominator. Both numerator and denominator have a factor of \$6$ and a variable $x$.

Factor out the common terms: $\frac{6 \times x}{12 \times x}$ can be rewritten as $\frac{6}{12} \times \frac{x}{x}$.

Simplify the fraction $\frac{6}{12}$ by dividing numerator and denominator by their greatest common divisor, which is 6.

Since $\frac{x}{x} = 1$ (for $x \neq 0$), multiply the simplified fraction by 1 to get the final simplified expression.

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Struggling with Intermediate Algebra?

Simplify the rational expressions below:6x12x\frac{6x}{12x}

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Simplify the rational expressions below:
$6 x over 12 x$