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.

## 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?

Suppose you’re working on a problem and the quantity 8^{2} comes up. If you have to pick up the calculator to figure out that 8^{2} = (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.

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.

## 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,

Often students will get confused when negative numbers are involved. Here’s a handy chart to help you out.

## Logarithms

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!

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:

log_{2} 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

log_{2} 1024 = 10.

Well, what if the number did not show up in the list? For example, what about log_{2} 997 ? Well, that’s when your calculator can come to the rescue. Nevertheless, you could still reason that the value of log_{2} 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

## Conclusion

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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##### About 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!

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