Because of their clarity and concision, the laws of exponents lend themselves well to GMAT math, especially to the Data Sufficiency format. If math isn't your thing, then perhaps the last time you gave any thought to exponents was back in Algebra Two, and perhaps exponents weren't your favorites there either. Take heart! In this post, I will explain the properties you need to know to be successful on the GMAT Quantitative section.

Fundamentally, an exponent is how many times you multiply a number, that is, how many factors of a number you have. This is the fundamental definition. The expression **base**", **exponent**", and **power**." The action of raising something to an exponent is called "**exponentiation**."

Just as multiplication distributes over addition & subtraction

so exponentiation distributes over multiplication and division.

Why is that? Well, consider

All the laws of exponents make sense if you just go back to the fundamental definition.

In this context, I will say: beware of one of the most tempting mistakes in all of mathematics. Exponentiation does NOT distribute over addition.

Beware. Even when you know this is wrong, even when you make an effort to remember that it's wrong, the inherent pattern-matching machinery of your brain will automatically pull your mind back in the direction of making this mistake. You must be vigilant to avoid this mistake.

What happens when you multiply two unequal powers of the same base?

Well, let's think about a concrete example. Suppose we are multiplying

If we start with five factors of x, and "stir in" three more factors of x, we wind up with a total of eight factors. All we have to do is add the exponents. We can simply generalize this pattern:

Don't just memorize this: make sure you understand the logic that leads to it. Remembering with the logic is 100x more effective than blind memorization!

What happens when you divide two unequal powers of the same base?

As with last time, a concrete example will illuminate the question. Suppose we divide

If we start out with seven factors, and then cancel away three of them, we are left with four. We just subtract the exponents. We can also generalize this pattern:

Once again: understand the logic, because remember through understanding is considerably more powerful than blind memorization.

Once again: understand the logic, because remember through understanding is considerably more powerful than blind memorization.

Mathematicians love to extend patterns. One example of this is the zero exponent. If we just see

One clever trick we can us is to employ the pattern found in division of powers. Suppose we have

but just fundamental logic would tell us that anything over itself equals one. Therefore, this expression

Here, we will extend the patterns even further. Consider this chart, for a base of

Exponent | |||||

Power |

Each time we move one cell to the right, the power gets multiplied by

Exponent | |||||||||

Power |

All we have done was to extend the pattern "**move one cell to the left, and the power gets divided by 2**." The result, we see, is that negative powers are reciprocals of their corresponding positive powers. This is consistent with the Division of Powers rule: if dividing means subtract the exponents, then an exponent of

I can't urge enough: the key to remembering these rules is understanding the logic of the arguments behind them. If you understand these rules, you will understand whatever the GMAT throws at you concerning exponents.

**Q1**. If

A.

B.

C.

D.

E.

Explanation:

To solve this problem, it is very helpful to express all of the quantities in terms of the same base. Once we do that, we can make use of the various Laws of Exponents to simplify the quantities further.

First, we'll express all of the quantities in terms of the same base of

We can now simplify all the powers that are being raised to a power via this Law of Exponents:

Next, we can simplify the numerator via this Law of Exponents:

We can then simplify the entire fraction via this Law of Exponents:

Lastly, we'll use the following rule to solve for

"If two powers with the same base are equal, then the exponents must be equal. That is, if

**Answer = A**

**Q2**. If

A.

B.

C.

D.

E.

**Answer = E**

**Q3**. If x and y are positive odd integers, then which of the following must also be an odd integer?

I.

II. x(y+1)

III.

A. I only

B. II only

C. III only

D. I and III

E. None of the above

Explanation:

**Case I**: Since

This case is acceptable.

**Case II**:

This case doesn't work.

**Case III**: The portion

There are actually two different possibilities that arise from this setup. For most cases, we'll get an odd number in the end. For example:

etc.

However, zero is also an even number, and anything raised to the power of zero results in

etc.

Since we can end up with both even and odd numbers from this case, this case doesn't work.

Only **Case I** guarantees an odd integer.

**Answer = A**