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## SAT

### Course: SAT > Unit 10

Lesson 2: Passport to advanced mathematics- Solving quadratic equations — Basic example
- Solving quadratic equations — Harder example
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems — Harder example
- Manipulating quadratic and exponential expressions — Basic example
- Manipulating quadratic and exponential expressions — Harder example
- Radicals and rational exponents — Basic example
- Radicals and rational exponents — Harder example
- Radical and rational equations — Basic example
- Radical and rational equations — Harder example
- Operations with rational expressions — Basic example
- Operations with rational expressions — Harder example
- Operations with polynomials — Basic example
- Operations with polynomials — Harder example
- Polynomial factors and graphs — Basic example
- Polynomial factors and graphs — Harder example
- Nonlinear equation graphs — Basic example
- Nonlinear equation graphs — Harder example
- Linear and quadratic systems — Basic example
- Linear and quadratic systems — Harder example
- Structure in expressions — Basic example
- Structure in expressions — Harder example
- Isolating quantities — Basic example
- Isolating quantities — Harder example
- Function notation — Basic example
- Function notation — Harder example

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# Operations with rational expressions — Basic example

Watch Sal work through a basic Operations with rational expressions problem.

## Want to join the conversation?

- You put the wrong exponent for a when you was explaining @3:15. It's a^3-a which gives you a^2(5 votes)
- Yes, you're right. And if you look carefully, the answer says 5a^2, not 5a as Sal concluded in the end.(6 votes)

- Can we still solve the problem without flipping the denominator and multiplying it with the numerator?(3 votes)
- Hey! noticing Sal says, "If you were in timed setting you might want to cut to the chase." a lot in these videos when he is writing the steps down. How could I cut "cut to the chase" if I find I wont be able to correctly answer the question if I do not write down all my steps? I want to retake my SAT sometime this year and fear heavily I will not have enough time given how a lot of the questions I have to write down very similarly to how Sal does things.(1 vote)
- In these videos, Sal takes his time through the problems, explaining as he goes. On the actual test, you might take shortcuts or do things slightly messier to get the answer faster.

For example, for this problem you might be familiar enough with how SAT questions work to understand that x^2 + 5x - 24*must*have a factor of (x-3), just based on the fact that you need to make a common denominator. You would then be able to skip coming up with two factors, and instead just find the matching factor to x-3 that multiplies to the -24 and then check really quickly to make sure you didn't assume wrongly. It goes without saying that you would also show much less work, like taking out the arrows for distribution and directly going from 3 - 7(x+8) to 3 - 7x - 56, for example.

All of the speeding up parts of studying for the SAT generally come after you're confident about every question type and you start to recognize the patterns of the test. If you keep practicing enough, subtracting rational expressions like these will seem like muscle memory, at which point you can start to cut out steps of Sal's process as you see fit.(5 votes)

- what is a rational expression?(1 vote)
- Rational refers to a ratio, which can be written as a division expression. Technically anything with a division sign has a rational expression, but in my experience the term is usually reserved for when you have variables in the denominator of an expression that you have to solve.(3 votes)

- why would you put a2b3/5c(0 votes)
- Are you asking why the answer is not a²b³/5c ?

I am guessing that you did not flip the denominator completely when solving this...

So, we start with a fraction consisting of one fraction in the numerator and one fraction in the denominator; it is a quotient of fractions.

Let's look at a really simple example of 4/2 which is equal to 2

We can also think of this as 4 times ½, (which is the reciprocal or inverse of 2)

4 times ½ = 2, of course

Now returning to fractions, if we look at a simpler example without all those variables:`½ / ⅜`

we can do the same thing as we just did. We remember that dividing by a fraction is the same as multiplying by the denominator's inverse.

So, we can say ½ / ⅜ → ½ ÷ ⅜ → ½ ∙ 8/3

Now we can multiply the numerators to get

1 ∙ 8 = 8 and we can multiply the denominators to get 2 ∙ 3 = 6 as our new denominator

8/6 can simplify to 4/3

Now let's look at our SAT practice question:`3a³b⁵/c³ is our numerator`

. We cannot simplify this fraction (rational expression) any further`15ab²/25c⁴ is our denominator`

. We**could**simplify this expression further if we want, but we can do this later also. The most important thing to do is to remember to**take the reciprocal**of this denominator (flip the fraction over) and then multiply the reciprocal times our numerator:`3a³b⁵/c³ is our numerator`

.`25c⁴/15ab² is our flipped fraction`

.`3a³b⁵/c³`

∙`25c⁴/15ab²`

If we focus on our numerical values and**ignore the variables**, we have`3`

∙`25/15`

or`3 ∙ 25/15`

=`75/15`

If we add the variables back in, we have`75/15`

∙`a³b⁵/c³`

∙`c⁴/ab²`

This simplifies to 75a²b³c/15 or`5a²b³c`

Hope that helps(9 votes)

- Appreciate the free informative lessons <3(1 vote)
- You put the wrong exponent for a when you was explaining @3:15. It's a^3-a which gives you a^2(1 vote)
- When you made 15ab^2/25c^4 its opposite reciprocal, why didn't the exponents(or even numbers) become negative? When switching exponents from numerator to denominator(and vice-versa) the exponent usually becomes negative.(0 votes)
- First, when you take the reciprocal, numbers don't become negative. (They change when you take the
*negative*reciprocal)

The exponent's sign changes only if the base stays in its place. Here's an example:

you have x^2, the reciprocal could be either x^-2 OR 1/x^2, which mean the same thing.

for x^2/y^3, its reciprocal could be x^-2/y^-3 or simply y^3/x^2.

So as a rule of thumb, if the base changes its place (e.g. from the numerator to the denominator and vice versa), then the exponent doesn't change its sign. On the other hand, if the base stays where it is, only the exponent's sign should be changed.

Hope it did clear some things for you.(5 votes)

- At1:44, why did he switch the 25c4?(0 votes)

## Video transcript

- [Instructor] Which
expression is equivalent to the above difference? All right, this is gonna be fun. So we're adding these two, I guess we could say rational expressions. And whenever you add any
expressions like this, if you add any fractions, you would wanna have a common denominator. And the same thing is true over here. And so one thing we can do is
let me actually see if I can, actually, let me rewrite the whole thing. So three over x squared
plus five x minus 24 minus seven over x minus three. Now, instead of, I guess,
the brute force way is to just multiply the denominators and say, okay, that's gonna be a common multiple. But there might be a
cleaner way of doing that, especially if x minus three
is one of the factors of this. So let's see if this is. So let's see, negative
three times positive eight is gonna be negative 24. And the negative three plus
positive eight is positive five. So this actually can be factored
into x plus eight times, x plus eight times x minus three. This is what this thing is and if this step looks unfamiliar to you, I encourage you to watch the video on factoring quadratic expressions. But what's neat about this is now we say, okay, look, to have the
same denominator here, we just have to take this fraction and multiply the numerator and
denominator by x plus eight. So if we multiply the
denominator by x plus eight, then these two denominators
are equivalent. But I can't just multiply
only the denominator by x plus eight. I also have to multiply the
numerator times x plus eight. And so what does this simplify to? This is going to be equal to three over x plus eight times x minus three. Times x minus three. And then minus, and actually, let me distribute the seven. Minus, remember this
negative sign out front, so it's gonna be minus seven. Actually, let me just
write it this way first. Minus seven x plus 56 over x plus eight times x minus three. If I was really under
time pressure on the SAT, I wouldn't probably do all of these steps. I would try to get to the chase, cut to the chase a little bit faster. But this is hopefully so it
helps us understand things. And so this is going to be equal to. We have our common denominator. X plus eight times x minus three. So it's going to be three. And we can distribute this negative sign. Minus seven x minus 56. So it's going to be equal to, let's see, we can write negative seven x. And then three minus 56 is
minus 53 or negative 53. And x plus eight times x minus three, we already established, is the same thing as x squared plus five x minus 24. And that is exactly, that's exactly this
choice right over there.