AP Calculus Exam: Arithmetic Review

It may seem strange to talk about arithmetic when you’re prepping for an AP Calculus exam. However, in order to succeed on the test, it’s important to have all the basics down: algebra, geometry, and yes, even arithmetic.

Now keep in mind, we’re not just talking about adding and subtracting here. The term arithmetic applies to a wide range of techniques for manipulating numbers and using number properties.

AP Calculus Exam Arithmetic Review -magoosh

Arithmetic on the AP Calculus Exam

As you may already know, both the Calculus AB and BC tests have sections that require a calculator and sections that do not allow them. (A more detailed description of the tests along with helpful advice can be found here: How Long is the AP Calculus Exam?)

To Calculate or Not To Calculate: That is the Question

Obviously, you’ll have to know your numbers on the no-calculator sections, but did you realize how mathematical fluency can help you throughout the exam?

calculus and calculator

Try not to go to the calculator for easy calculations.

Suppose you’re working on a problem and the quantity 82 comes up. If you have to pick up the calculator to figure out that 82 = (8)(8) = 64, then you’ve just lost 5 seconds.

Now that may not sound like much, but considering that each multiple choice problem should take no more than 2 minutes, those 5 seconds really do make a difference!

Dealing with Fractions

It goes without saying that you should be able to add, subtract, multiply, and divide numbers. Most AP calculus students have no difficulty with these operations… except when it comes to fractions.

The rules for working with fractions are just a little more involved.

For example, when adding or subtracting fractions, you must have a common denominator. For instance, 2/5 + 7/8 is NOT equal to 9/13. Instead,
find the common denominator, 5 × 8 = 40, and re-express each fraction.

2/5 + 7/8 = 51/40

Remember, multiplication of fractions is fairly easy: just multiply across the top and across the bottom. Dividing by a fraction involves taking the reciprocal of the bottom fraction and then multiplying.

Formulas for multiplication and division of fractions

Powers and Roots

Another area that seems to cause problems is in working with powers and roots.

Remember that a power (or exponent) stands for repeated multiplication, and a root is the inverse operation for a power. For example,
Fourth root of 81 is 3.

Often students will get confused when negative numbers are involved. Here’s a handy chart to help you out.Chart showing powers and roots of negative quantities


Finally, let’s talk logarithms. You may say that logs are part of algebra, not arithmetic, but I prefer to think of them as just another operation that you can do with numbers. Plus, they’re just really cool!

Nautilus - logarithmic spiral

Logarithms can be beautiful! The nautilus shell grows in the shape of a logarithmic spiral.

A logarithm allows you to isolate a power. For example, if you wanted to know what power of two would give you 1024, then you could phrase it as a logarithm problem:

log2 1024 = ?

I bet you could figure this one out. Count up the powers of two: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Aha! The tenth number in the list is exactly 1024, so

log2 1024 = 10.

Well, what if the number did not show up in the list? For example, what about log2 997 ? Well, that’s when your calculator can come to the rescue. Nevertheless, you could still reason that the value of log2 997 is something between 9 and 10 without even touching your calculator!

Of course, there’s much more to know about logarithms. Check out the following link for a good review: AP Calculus Review: Properties of Exponents and Roots


It’s essential to know your arithmetic for the AP Calculus exams, whether working on a section that allows a calculator or not. Review fractions, exponents, and logarithms especially, as these topics seems to give us the most trouble.

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  • Shaun Ault

    Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!

By the way, Magoosh can help you study for both the SAT and ACT exams. Click here to learn more!

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