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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > 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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# Interpreting nonlinear expressions — Harder example

Watch Sal work through a harder Interpreting nonlinear expressions problem.

## Want to join the conversation?

- I have my SAT tomorrow and I am reviewing some math. How is 64 distributed. I know you explained this in a previous comment, but if you look closely, d is not being distributed. Therefore, if you are multiplying d by 64, why is it necessary to distribute every number by 64 if it was not original apart of the distributive property.(25 votes)
- If 4d^3 = (4^3)(d^3), then it amounts to 64d^3. 64 is distributed because the equation is 3.98*10^-20*64d^3. All terms are being multiplied by 64, so it can be factored. Sorry that your question wasn't answered until now! :((11 votes)

- Doesn't factoring out the 64 imply that the the 64 is multiplied by every term in the parentheses?(18 votes)
- You don't have to do this in this specific case because we just have to see how many times longer Mar's revolution time would be compared to that of Mercury's. We just have to see that one value which shows how much longer the revolution would take (8). Multiplying 64 with every single term in the parentheses (after finding the principal root which is done near2:50) would just add an extra step. Remember, the SAT is time constrained, so the quickest option should be used!(8 votes)

- Isn't 6% 0.06? Then why did you consider that it is 1.06 in Interpreting nonlinear expressions — Harder example?(9 votes)
- 6% is indeed .06, but that's not all the question asks you for. Instead, the answer choices describe "an increase of 6%". To put that into math, it's the original price plus 6% of what it was:

x + .06x = x(1.06)

That's how we get 1.06 for the 6% increase, and why it's in there instead of plain .06. If the function was 10000(.06)^(q/4), it would mean that the styling service only retains 6% of its subscribers per year, or that it loses 94% every year.(16 votes)

- Hi! I have a question... I don't understand how you would square root the whole equation after you substitute the number 4 into d (for distance)? Can someone please explain it to me? Thank you in advance! xx(3 votes)
- Can anybody explain me that q/4 how its not 1 quarter? I am utterly confused(7 votes)
- If we have q = 1 it means it is one quarter. When it is q = 4 its is one whole year as in one year we have 4 quarter(6 votes)

- For those who do not understand why q/4 is not 4q.

First, q/4--

We all know that q is a quarter year. Right? In 4(1.01)^q/4, the reason why we want to make q/4 is that we want to make the degree of 1.01 one. Meaning, 4 quarters are equal to 1 year. So the degree is 1 and 4(1.01) ^4/4 = in one year the number is increasing by 1 percent.

Second, if we do q*4 then--

4(1.01)^4q imagine we want to find the yearly increase. What do find here then? Okay 1 year=4 quarters, put it. 4(1.01)^4*4. See the degree is not one.

Well, additionally. We need 4 circles. We cannot do it by 4q but q/4.(7 votes) - At2:12when 1000(1.06) and 1000 is increasing by (1.06). Why is it increasing by 6%, and not 106%? Where does the 1 away from (1.06) go?(3 votes)
- 1.06 -> 106%

106% - 100% = 6% > 1 (increasing)(5 votes)

- Is there any other way to solve this problem without having to make a T-chart?(4 votes)
- Hi Rachel!

I think the best way is elimination. Since we know we have an exponent in the equation, we know we have an exponential function which means that it's not increasing by a certain amount each year but rather by a certain factor. (Eliminate a and b). To increase by 6% we need to make sure that 1.06 is only raised to the first power. That means q must equal 4 since 4/4 = 1. q equaling 4 means that that's four quarter years. What's four quarter years? 1 year.

I know that maybe looked long and hard, but when you do it in real life, it takes much less time than Sal's way.(4 votes)

- What kind of topics are there in sat(3 votes)
- The Math SAT has 3 major categories: Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math.

See the CollegeBoard website for a more detailed explanation.

https://collegereadiness.collegeboard.org/sat/inside-the-test/math

Sal covers all these topics in the videos provided.(4 votes)

- It says 4 times as distant from the sun as Mercury is, and the distance is d^3. Why is (4d)^3 instead of 4(d^3)?(4 votes)
- I'm not sure, but since the distance of Mercury is represented by d, you multiply 4 by d, and also by the exponent. The exponent remains 3.(3 votes)

## Video transcript

- [Instructors] We're told
the equation above models the number of subscribers, S,
of an online styling service q quarter years after
the service launches. Based on the model, which
of the following statements is true. And let's see, they have
a bunch of statements on how much the subscribers increased by either each quarter or each year. And these first two, they're
talking about increasing about fixed amount each quarter a year. And here it's a percentage
each quarter or year. So let's look at this expression. And the immediate thing that
probably jumps out of you is that we have an exponent here. This is an exponential expression. And so let's actually just
make a little table here to think about how does S grow with Q and what I'm going to do. So this is Q and this is S and I'm going to pick some Q values that'll make the math a little bit simple 'cause we noticed that the
exponents here is Q over four. So let's say what happens
at when Q is equal to zero 'cause zero over four is
pretty easy to calculate. It's not going to be a fraction. Let's say when only four
quarters have gone by 'cause four over four is pretty easy. And maybe when eight quarters have gone by which is the same thing as two years. So when Q is equal to zero, this is going to be
1.06 to the zero power, which is this one. So we have 1000 subscribers. And so you could view it as that's the initial
amount of subscribers. Then when Q is equal to
four, what do we have? Well, this exponents is
going to be four over four. So it's just going to be one. So it's going to be 1000, 1000 times 1.06. And then when Q is equal to eight, eight divided by four is two. So it's going to be 1.06
to the second power. So that's going to be 1000. We could write it times 1.06, actually let me just write it this way. Times 1.06 to the second power. So let's think about how it's growing at least every four quarters which is the same thing as one year. Well, it looks like we are growing or we are multiplying by 1.06 or another way of thinking about it is we are increasing by 6%. Once again, when we go from
quarter four to quarter eight which is another year goes by, we're multiplying by another 1.06. So that's growing by another 6%. So it looks like we're growing by 6% every four quarters or every year. And so the choice that does that, the number of subscribers
increases by 6% each year, that one looks good. Now let's think about the other
choices, just to feel super. If we're doing this on a test,
we would just keep going. But just to feel really confident that it's not these other choices. The number of subscribers
increases by 60 each quarter year. Well, this would be a linear equation. It would look something
like that, like this. S is equal to your initial
number of subscribers plus the number of quarter years times 60, it would look something
like this for choice A which clearly we don't have over here. So we can rule that out. The number of subscribers
increases by 60 each year. Well, this would be
something like subscribers is equal to 1000, our
initial number of subscribers plus Q over four times 60 because then every four quarters you would increase by a year, and obviously you could simplify this. It would be 15 times Q but this is clearly not
what is happening here. And then the number of subscribers increases by 6% each quarter year. Well, the exponential
that would describe that would be our initial number
of subscribers times 1.06, 1.06 not to the Q over four,
that would just be to the q so that every time we
are a new quarter year or every time a quarter goes
by, you would increase by 6% but here we clearly have Q over four. So every four quarters
we're increasing by 6%.