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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 @. It's a^3-a which gives you a^2
• Yes, you're right. And if you look carefully, the answer says 5a^2, not 5a as Sal concluded in the end.
• Can we still solve the problem without flipping the denominator and multiplying it with the numerator?
• 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.
• 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.
• why would you put 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
• Appreciate the free informative lessons <3
(1 vote)
• Easy!!
(1 vote)
• You put the wrong exponent for a when you was explaining @ . 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.
• 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.