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GMAT Exponents and Roots

by ejkiv


July 18, 2015


GMAT Exponents

There are a variety of different rules that you are going to have to memorize on exponents.  There is no way around this, as the GMAT will find ways to test your knowledge of exponents from all different angles.  We will work on giving you the logic as well so that you do understand these rules and move beyond pure memorization.

The basic anatomy of an exponent is as follows:

In this example, x is called the base and y is called the exponent.

Another example is as follows:

Here, we have added a 2 in front of the base.  This 2 is call the coefficient.

We will work with these later, but it is important to have the terminology down now.  First, an exponent tells you how many times a number should be multiplied by itself.

Each of the individual 3’s above could be written as 3^1, as a number with no exponent is the same as having and exponent of 1.  Also, any number raised to the zero power equals one.  Thus:

 

Now, we will go through the different operations with exponents with both numbers and variables so that you better understand how to use them.

Multiplying and Dividing Bases

We are starting with multiplication and division because they are skills that you will have to know in order to do addition and subtraction.  We talk in terms of bases because the bases will never change value unless the exponents are identical.  In other words, .  We will talk about the exception in the factoring bases section.

Using the first example, if we work backwards it is easy to see that you can combine like bases.  This is done by simply adding the exponents.  Also, when you are multiplying bases with coefficients, you need to multiply the coefficient as well.

All of the rules for division are the same as multiplication; however, you simply subtract the exponents when dividing rather than adding when you multiply. Here are some examples:

Adding and Subtracting Bases

There are two different ways in which you will add or subtract bases. The first is by simply adding and subtracting the coefficients of identical terms. The second is by factoring - we will use this to explain why the first method works.

Because this is important, I am going to rewrite this: In order to add and subtract bases you must have identical terms. Here are some examples:

Exponent with Bases that have Exponents

You will also encounter examples where you raise a base with an existing exponent to another exponent. In this case, you will multiply the exponents. It looks like this:

Exponent Behavior

 

GMAT ROOTS

Square Root: For the number y, the number such that x^2 is equal to y is the square root of y. In other words, 3 is the square root of 9 because 3^2 = 9.Cube Root: For the number y, the number such that x^3 is equal to y is the square root of y. In other words, 3 is the cube root of 27 because 3^3 = 27.It is important to know the relationship of roots and exponents.  This is displayed here: Multiplying and Dividing Bases Remember, we talk in terms of bases because the bases will never change value unless the exponents are identical. To combine like bases, you must first convert the radical to an exponent as we did in the previous paragraph. Using the first example, if we work backwards it is easy to see that you can combine like bases. Similarly, you will have to multiply coefficients. All of the rules for division are the same as multiplication; however, you simply subtract the exponents when dividing rather than adding when you multiply. Adding and Subtracting Bases Adding and subtracting is also identical with roots. Here are a few examples: Roots with Bases that have Exponents or Roots You will also encounter examples where you raise a root to an exponent or another root. In this case you will multiply the exponents after conversion. It looks like this:
Other notes Roots have one thing that at face value seems a bit different, but when broken down into the rules we talked about with exponents they make perfect sense. Let’s look at a few examples: This also, works when dividing roots. The important thing is that you will have to be able to move back and forth when appropriate in order to simplify expressions. Root Behavior As with exponents, roots require us to test zones when looking at data sufficiency. Take note of the behavior of in these different areas. Another piece of information is that you cannot take the square root of a negative number because no product of the same number can give you a negative number. One last thing about roots. It is considered, improper form to leave a root in the denominator of a fraction. Not that you have to worry about what is proper or not, but you will find the correct answers will not include radicals in the denominator. You must remove them. Here are two different ways this will happen:

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