Chapter 6

Roots

This chapter discusses some of the ways roots are incorporated into expressions and equations and the ways you are allowed to manipulate them. You may be tempted to use the on-screen calculator when you see a root expression, but it's often much easier to go without. You just need to know your roots rules!

Before getting into some of the more complicated rules, it is important to remember that any square root times itself will equal whatever is inside the square root, for instance: , . You can even apply this rule to variables: . So the first rule for roots is:

Multiplication and Division of Roots

Suppose you were to see the equation , and you were asked to solve for x. What would you do? Well, , so you could rewrite the equation as 3 + 2 = x, and you would know that x = 5. Because 4 is a perfect square, you were able to simply evaluate the root, and continue to solve the problem. But what if the equation were , and you were asked to find x? What would you do then? Neither 8 nor 2 is a perfect square, so you can't easily find a value for either root.

It is important to realize that, on the GRE, sometimes you will be able to evaluate roots (when asked to take the square root of a perfect square or the cube root of a perfect cube), but other times it will be necessary to manipulate the roots. Up next is a discussion of the different ways that you are allowed to manipulate roots, followed by some examples of how these manipulations may help you arrive at a correct answer on GRE questions involving roots.

Go back to the previous question: If , what is x?

When two roots are multiplied by each other, you can do the multiplication within a single root. What that means is that you can rewrite as , which equals . And equals 4, which means that x = 4.

This property also works for division.

If , what is x?

You can divide the numbers inside the square roots and put them inside one square root. So becomes , which becomes . And equals 3, so x = 3.

Note that these rules apply if there are any number of roots being multiplied or divided. These rules can also be combined with each other. For instance, becomes . The numbers inside can be combined, and ultimately you end up with , which equals 6.

Check Your Skills

Solve for x.

1. x =

2. x =

3. x =

4. x =

Answers can be found on page 123.

Simplifying Roots

Just as multiple roots can be combined to create one root, you can also take one root and break it apart into multiple roots. You may be asking, why would you ever want to do that? Well, suppose a question asked: If , what is x? You would combine them, and say that x equals . Unfortunately, will never be a correct answer on the GRE. The reason is that can be simplified, and correct answers on the GRE are presented in their simplest forms. So now the question becomes, how can you simplify ?

What if you were to rewrite as ? As mentioned, you could also break this apart into two separate roots that are multiplied together, namely And you already know that equals 2, so you could simplify to . And in fact, that is the simplified form of , and could potentially appear as the correct answer to a question on the GRE. Just to recap, the progression of simplifying was as follows:

Now the question becomes, how can you simplify any square root? What if you don't notice that 12 equals 4 times 3, and 4 is a perfect square? Amazingly enough, the method for simplifying square roots will involve something you're probably quite comfortable with at this point—prime factorizations.

Take a look at the prime factorization of 12. The prime factorization of 12 is 2 × 2 × 3. So can be rewritten as . Recall the first roots rule—any root times itself will equal the number inside. If can be rewritten as , you can take that one step further and say it is . And you know that .

You can generalize from this example and say that when you take the prime factorization of a number inside a square root, any prime factor that you can pair off can effectively be brought out of the square root. Try another example to practice applying this concept. What is the simplified form of ? Start by taking the prime factorization of 360:

360 = 2 × 2 × 2 × 3 × 3 × 5

Again, you are looking for primes that you can pair off and ultimately remove from the square root. In this case, you have a pair of 2’s and a pair of 3’s, so you can separate them:

Notice that the prime factorization of 360 included three 2’s. Two 2’s could be paired off, but that still left one 2 without a partner, therefore represents the prime factors that cannot be paired off. This expression can now be simplified to , which is .

You might have seen right away that 360 = 36 × 10, so . The advantage of the prime factor method is that it will always work, even when you don't spot a shortcut.

Check Your Skills

Simplify the following roots.

5.

6.

7.

Answers can be found on page 123.

Solving Algebraic Equations Involving Exponential Terms

GRE exponent problems sometimes give you an equation, and ask you to solve for either an unknown base or an unknown exponent.

Unknown Base

The key to solving algebraic expressions with an unknown base is to make use of the fact that exponents and roots are inverses, just as multiplication and division are, and so can be used to effectively cancel each other out. In the equation x³ = 8, x is raised to the third power, so to eliminate the exponent you can take the cube root of both sides of the equation.

This process also works in reverse. If you are presented with the equation , you can eliminate the square root by squaring both sides. Square root and squaring cancel each other out in the same way that cube root and raising something to the third power cancel each other out. So to solve this equation, you can square both sides and get , which can be simplified to x = 36.

There is one additional danger. Remember that when solving an equation where a variable has been squared, you should be on the lookout for two solutions. To solve for y in the equation y² = 100, you need to remember that y can equal either 10 OR −10.

 

Unknown Base	Unknown Exponent
x3 = 8	2x = 8

 

Check Your Skills

Solve the following equations.

8. x³ = 64

9.

10. x² = 121

Answers can be found on page 123.

Unknown Exponent

Unlike examples in the previous section, you can't make use of the relationship between exponents and roots to help solve for the variable in the equation 2^x = 8. Instead, the key is to once again recognize that 8 is equivalent to 2³, and rewrite the equation so that you have the same base on both sides of the equal sign. If you replace 8 with its equivalent value, the equation becomes 2^x = 2³.

Now that you have the same base on both sides of the equation, there is only one way for the value of the expression on the left side of the equation to equal the value of the expression on the right side of the equation—the exponents must be equal. You can effectively ignore the bases and set the exponents equal to each other. You now know that x = 3.

By the way, when you see the expression 2^x, always call it “two TO THE xth power” or “two TO THE x.” Never call it “two x.” “Two x” is 2x, or 2 times x, which is simply a different expression. Don't get lazy with names; that's how you can confuse one expression for another.

The process of finding the same base on each side of the equation can be applied to more complicated exponents as well. Take a look at the equation 3^{x + 2} = 27. Once again, you must first rewrite one of the bases so that the bases are the same on both sides of the equation. Because 27 is equivalent to 3³, the equation can be rewritten as 3^{x + 2} = 3³. You can now ignore the bases (because they are the same) and set the exponents equal to each other: x + 2 = 3, which means that x = 1.

Check Your Skills

Solve for x in the following equations.

11. 2^x = 64

12. 7 ^{x − 2} = 49

13. 5^3x = 125

Answers can be found on page 123.

 

Check Your Skills Answer Key

1.   10:

2.   2:

3.   12:

4.   8:

5.   

6.   

7.   21:

8.
      

9.
      

10.
       

11. 6: 2^x = 64

2^x = 2⁶

x = 6

12. 4: 7^x−2 = 49

7^x−2 = 7²

x−2 = 2

x = 4

13. 1: 5^3x = 125

5^3x = 5³

3x = 3

x = 1

 

Problem Set

1.

Quantity A		Quantity B
		12

2.

36< x <49

Quantity A		Quantity B
		43

3.

Quantity A		Quantity B

 

Solutions

1. (A): One of the root rules is that when two individual roots are multiplied together, you can carry out that multiplication under a single root sign:

While this can be simplified (,) you're actually better off leaving it as is:

Quantity A		Quantity B
		12

Now square both quantities:

Quantity A		Quantity B
		(12)2 = 144

Therefore, Quantity A is greater.

2. (A): The common information tells you that x is between 36 and 49, which means the square root of x must be between 6 and 7. Rewrite Quantity A:

36 < x < 49

Quantity A		Quantity B
		43

Now rewrite Quantity B so that it has a base of 2 instead of a base of 4:

36 < x < 49

Quantity A		Quantity B
		43 = (22)3 = 26

The value in Quantity A must be greater than 2⁶, and so must be greater than the value in Quantity B. Therefore, Quantity A is greater.

3. (B): Simplify both quantities by combining the roots into one root.

Quantity A		Quantity B

Now simplify the fractions underneath each root.

Quantity A		Quantity B

Because is larger than , Quantity B is greater.