Divide each expression and write the quotient in its simplest form. y − 1 y + 2 y − 1 3 ( y + 2 ) \frac{y-1}{\frac{y+2}{\frac{y-1}{3\left(y+2\right)}}}

we are asked to divide each expression and write the quotient in its simplest form. So we have y minus one over y plus two, and that's all over y minus one over y, or over three times y plus two. So how do we solve a problem like this? Well, the thing that you need to realize is these division bars, really any of these bars that you see, are just division. Now we don't typically write that division form for a single fraction like this. However, we can do it when there's two fractions that are being divided. So I can also write this as y minus one over y plus two divided by and that's gonna be y minus one over three times y plus two. Based on the problems we've been solving, this just might make it look a little bit more clear. Now, we can still deal with these the exact same way. We can use keep, change, and flip. So if I wanna use keep, change, flip, y minus one over y plus two stays the same. The division changes to multiplication, and then we flip the second fraction. So we get three times y plus two over y minus one. Now notice when writing everything like this, well, what I can see is that the y minus ones, well, those are actually going to cancel. So when we multiply everything across, I can just cancel the y minus ones, and then I can also see that the y plus twos are going to cancel also. So really, all we're going to end up with is three left over. Now, if you wanted to write this all out and see for yourself, you could. So if you wanted to combine this all into one fraction, so like we've done in the past, and then we end up with y plus two times y minus one, this is perfectly okay to do also. Because notice when we go ahead and write everything as one fraction, the y minus ones and the y plus twos cancel anyways. So you can go about doing this a bit more quick, or you can combine everything into one, but Either way you go about this, you should end up with just three. So three ends up being the solution to this problem, and that's what happens when we simplify. So remember, the main thing that you need to remember is to keep, then change, and then flip. And that's exactly what we did over here, and you can choose to simplify in that step, or you can combine everything into one fraction and simplify from there. So I hope you found this video helpful, and I encourage you to continue checking out the examples and practice to get more practice with this concept and really make sure you have it committed to memory. See you all in the next one.

Divide each expression and write the quotient in its simplest form. y − 1 y + 2 y − 1 3 ( y + 2 ) \frac{y-1}{\frac{y+2}{\frac{y-1}{3\left(y+2\right)}}}

3 y − 1 \frac{3}{y-1} y − 1 3

8. Rational Expressions and Equations

Multiplying and Dividing Rational Expressions

Beginning & Intermediate Algebra

8. Rational Expressions and Equations

Multiplying and Dividing Rational Expressions

Struggling with Beginning & Intermediate Algebra?

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Multiple Choice

Divide each expression and write the quotient in its simplest form.
$\frac{y - 1}{\frac{y + 2}{\frac{y - 1}{3 (y + 2)}}}$

$\frac{1}{3}$

$\frac{3}{y + 2}$

$3$

$\frac{3}{y-1}$

Verified step by step guidance

Start by carefully examining the complex fraction: $\frac{y-1}{\frac{y+2}{\frac{y-1}{3(y+2)}}}$. Notice that the denominator itself is a fraction divided by another fraction, so we will simplify from the innermost fraction outward.

Focus on the innermost fraction in the denominator: $\frac{y-1}{3(y+2)}$. This fraction is the denominator of the next fraction up, which is $\frac{y+2}{\frac{y-1}{3(y+2)}}$. To simplify this, rewrite the division as multiplication by the reciprocal: $\frac{y+2}{\frac{y-1}{3(y+2)}} = (y+2) \times \frac{3(y+2)}{y-1}$.

Simplify the multiplication in the denominator: $(y+2) \times \frac{3(y+2)}{y-1} = \frac{3(y+2)^2}{y-1}$. Now the original expression becomes $\frac{y-1}{\frac{3(y+2)^2}{y-1}}$.

Rewrite the overall expression as a division of fractions: $\frac{y-1}{\frac{3(y+2)^2}{y-1}} = (y-1) \times \frac{y-1}{3(y+2)^2} = \frac{(y-1)^2}{3(y+2)^2}$.

Look for any common factors or terms that can be simplified further. Since $(y-1)^2$ and $(y+2)^2$ are distinct and cannot be simplified, the expression is now in its simplest form.

3:41m

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Struggling with Beginning & Intermediate Algebra?

Divide each expression and write the quotient in its simplest form.y−1y+2y−13(y+2)\(\frac{y-1}{\frac{y+2}{\frac{y-1}{3\left(y+2\right)}\)}}

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Divide each expression and write the quotient in its simplest form.
$\frac{y - 1}{\frac{y + 2}{\frac{y - 1}{3 (y + 2)}}}$