GRE Exam: A Simple Guide To Algebra - Exponents and Roots

Exponent Basics
In the term 53, five is the base and 3 is the exponent.3 comes out to 5 × 5 × 5. Both the exponent and base can be any real number (not just positive integers).
Most GRE questions dealing with exponents will require you to use a few simple rules and manipulate them in unorthodox situations. But before getting to these rules, let’s look at some other properties of exponents:

An even exponent always yields a positive result.
The base of an exponential expression can be positive or negative, but when the base is raised to an even exponent, the result will always be positive.

Consider:

The most obvious solution to the preceding equation is 4. If you substitute 4 for x, you arrive at 42 = 16, which is a true statement. But notice that –4 is also a solution for x! If you substitute −4 back into the equation, you arrive at (−4)2 = (−4)( −4) = 16, which is also true.

The previous example illustrates the following general principle: An even exponent will hide the sign of the base.

In other words, whether the base is positive or negative, when it is raised to an even power, the result will be positive. This is because an even number of negative factors will always cancel out to create a positive product.

The flip-side of this fact concerns odd exponents. An odd exponent will preserve the sign of the base.

For example, if x3 = −8, then there is just one solution for x: −2. Notice that 2 is not a solution for x because if you plug it back into the equation, you arrive at (2)3 = 8, not −8.

Base of 0, 1, and −1
- When a base of zero is raised to any power, the result is zero: 02 = 0
- When 1 is raised to any power, the result is 1: 110 = 1; 1−300 = 1
- When −1 is raised to an even power, the result is 1. When −1 is raised to an odd power, the result will be −1: −110 = 1; −1−303 = −1

Fractional Base
- When a positive proper fraction (a number between 0 and 1) is raised to a power, an interesting property results: the resulting value is less than the original base:

Compare the preceding to what happens when you raise an integer base to a power:

- When a fraction is raised to a power, the exponent distributes to the numerator and denominator of that fraction:

.

Exponent Rules
Most situations with exponents will require knowledge of basic exponent rules.

A good rule of thumb is that most exponent rules concern situations where either the base or the exponent is the same and where you’re either multiplying or dividing.

Multiplying Exponents with the Same Base: Add the Exponents
- When multiplying exponential terms with the same base, keep the base and add the exponents:

(35)(33) = 3(5+3). To understand why you are adding the exponents, write out 35 and 33. Notice that you arrive at: (3 × 3 × 3 × 3 × 3) (3 × 3 × 3). This leaves you with 3 multiplied by itself 8 times. Thus 38.

Dividing Exponents with the Same Base: Subtract the Exponents
- When dividing exponential terms with the same base, keep the base and subtract the exponents:

.

To understand why you are subtracting the exponents, write out 57 and 53.

Notice that you arrive at .

When you cancel out the common factors, you are left with four 5s in the numerator. Thus 54.

Raising a Power to a Power: Multiply the Exponents
- When raising an exponential term to a power, to simplify the term, you should multiply the exponents:

(54)3 = 5(4×3) = 512. Why? (54)3 = (54)(54)(54). This takes you back to the first scenario in which you were multiplying exponential terms with the same base. Recall that in such a situation, you should add the exponents. This will yield 5(4+4+4) = 512.

Multiplying and Dividing Exponents with Different Bases but the Same Exponent: Multiply or Divide the Bases
- When multiplying exponential terms with different bases but the same exponent, keep the exponent and multiply the bases.

Thus far, you have looked only at situations in which the base is the same. What about when the bases are different?

You can still manipulate the expression if the exponents are the same!

Why are you allowed to combine the bases? Again, write them out.

Notice that when you multiply these two terms, you will end up with 4 combinations of (5 × 3), giving you (5 × 3)4.

It is also important to notice that this rule works in reverse. If a product is raised to an exponent, then the exponent will distribute to all the factors in the product:

 

When dividing exponential terms with different bases but the same exponent, keep the exponent and divide the bases:

.

To understand why, once again expand the numerator and denominator.

You arrive at:

You thus have three times, giving you .

Negative Exponents: Flip the Base
When raising a number to a negative exponent, to get rid of the negative exponent, you simply flip the base:


Putting It All Together: Finding a Common Base
Knowing the preceding rules is essential for success on exponent questions, but the GRE will make such questions difficult by forcing you to evaluate expressions where it seems that none of these rules apply. To get past these difficulties, you should always be concerned with manipulating what’s given to you to get to the same base. By doing so, you can then use the rules that were just covered.

Look at the following: (84)(325) =
29
220
237
820
25620

To simplify the expression, rewrite the exponential terms to have the same base.

By doing so, you will be able to use the exponent rule that says you can add the exponents when multiplying exponential terms with the same base:
8 = 23, so 84 = (23)4. Using your exponent rules, you know this comes out to 212.
32 = 25, so 325 = (25)5. Using your exponent rules, you know this comes out to 225.

Now you can use your rules! The expression now reads: (212)(225). Since you are multiplying exponential terms with the same base, you keep the base and add the exponents: (212)(225) = 2(12+25) = 237. The correct answer is C.

Let’s look at another example:

7212
123
212
92
123

Again, your focus should be to manipulate the numerator and denominator so that all the terms are in their prime forms.
86 = (23)6 = 218
93 = (32)3 = 36
66 = (3 × 2)6 = 36 × 26

So your new fraction is:

Use your exponent rules for division and you get 2(18−6) × 30. And this comes out to 212. The correct answer is C.

Table 1 lists the major exponent rules.

You should commit these rules to memory and make sure that you understand the conceptual basis behind these rules.
 

Table 1 Exponent Rules Table

Table 2 lists common powers and roots that appear on the GRE.

Committing these rules to memory will help you save precious time on the exam.
 

Table 2 Common Powers & Roots

Solving for an Unknown Exponent
So far, most of the questions that you have looked at have concerned shortcuts for evaluating exponential expressions. Sometimes, however, you will be asked to solve for a variable that is in the place of an exponent.

Look at the following example:
If 2x = 8, then what is the value of x?

No exponent rule will work here. Instead, you must recognize the following property: If two values are equal, then they must have the same prime factorization.

To solve for x, you should thus rewrite 8 as the product of its prime factors. 8 = 23, so the equation now reads: 2x = 23.

Now that the bases are equal, you know the exponents are equal, so x = 3.
If x and y are integers, and (3x)(2y) = 324, what is the value of x + y?

SOLUTION: Rewrite 324 as the product of its prime factors:
324 = 9 × 36 = 3 × 3 × 6 × 6 = 3 × 3 × 3 × 2 × 3 × 2 = (34)(22).

Thus (3x)(2y) = (34)(22). Now that both sides of the equation are expressed in terms of the same bases, you know that x = 4 and y = 2.

Factoring Exponential Expressions
Sometimes you will be given an exponent question that concerns the addition or subtraction of exponential terms. Since exponent rules only apply to the multiplication or division of exponents, you should almost always factor when two exponential terms are added or subtracted. How will you do so?

Look at the following example:
232 – 230 is equivalent to which of the following?
22
223
210
230
2303

SOLUTION: 232 can be rewritten as (230)(22). You can thus rewrite the expression as (230)(22) – (230)(1). Since both terms share a factor of 230, the expression can be written as: 230(22 – 1) = 230(3). The correct answer is Choice E.

Roots
Roots are the opposite of exponents.

Generally, a root will be denoted using the following symbol:
If , then what is x?

SOLUTION: x is the positive number that when squared yields 16. 42 = 16, so the answer is 4.

Note that when determining the square root of a number, the answer will always be positive.

Even though (−4)2 = 16, −4 is not a solution for x in the question.

In the preceding example, 16 is a perfect square.

A perfect square is any number whose square root is an integer.

For example, 9 is a perfect square because its square root is 3, but 15 is not a perfect square, since its square root is 3.87. . . .

Multiplying and Dividing Roots
If you are asked to simply evaluate a square root, you can of course use your on-screen calculator.

But many root questions will require you to instead manipulate the root.

For example: .

When multiplying square roots, you can combine all the terms underneath one square root.

Thus

When dividing square roots, you can combine all the terms underneath one square root:

Note that these rules also work in reverse:


 

Simplifying Roots
Simplifying a perfect square root is straightforward and can always be done on your calculator.

But what if, after going through a question, you arrived at an answer of

If you looked at the choices, would not appear in any of them. Why?

Because is not simplified.

To simplify it, you have to take any perfect squares out of the radical.

You would simplify in the following way:

is the simplified form of .

Generally, when you are trying to simplify a square root, you should break it up into any known perfect squares and remove those perfect squares from the square root.

Example: Simplify.

SOLUTION: First, rewrite 150 as 25 × 6. Thus .

Now, break up the square root: .

Finally, simplify any perfect squares. , so the answer is .
 

Quantitative Comparison Strategy: Exponents
When solving Quantitative Comparison questions that test exponents, it is essential to recall both the rules discussed previously, and just as importantly, the exceptions to these rules.

Remember that success on Quantitative Comparison questions often requires testing for Choice D, which requires challenging your assumptions.

When solving a Quantitative Comparison question where either the exponent or the base is a variable, always test 0, 1, and −1.

For example:

If x is positive, then Column B will always be greater. But remember that the exponents 0, 1, and −1 have interesting properties, so make sure to test those cases to see whether Column B is always greater.
- If x = 0, then the two quantities are equal.
- If x = 1, then Column B is greater.
- If x = −1, then Column A becomes and Column B becomes , meaning that Column A is greater.

These different values give you contradictory relationships, and the answer is therefore D.

When comparing numbers expressed as exponents, make sure to express each column using the same base:

Rewrite 1,000 using a base of 10: 1,000 = 103.

Substitute 103 for 1,000 in Column A, and compare the new expressions:

Use your exponent rules for Column A: (103)600 = 101,800.

The two columns are equal, and the answer is therefore Choice C.