# Roots

*n*th root of

*b*, where

*n*is called the index,

*b*is called the base, and is called the radical. denotes that number which raised to the

*n*th power yields

*b*. In other words,

*a*is the

*n*th root of

*b*if .

^{2}= 9, and because (–2)

^{3}= –8. Even roots occur in pairs: both a positive root and a negative root.

^{4}= 16, and since (–2)

^{4}= 16. Odd roots occur alone and have the same sign as the base: since (–3)

^{3}= –27. If given an even root, you are to assume it is the positive root. However, if you introduce even roots by solving an equation, then you

__must__consider both the positive and negative roots:

*x*= ±3

There are only two rules for roots that you need to know for the test:

For example,

For example,

Caution: . For example, .

Also, . This common mistake occurs because it is similar to the following valid property: (If *x* + *y* can be negative, then it must be written with the absolute value symbol: ).

**Note,**in the valid formula, it’s the whole term,

*x*+

*y*, that is squared, not the individual

*x*and

*y*.

In this case, the roots can be added because both the indices and bases are the same. Sometimes radicals with different bases can actually be added once they have been simplified to look alike.

*x*

^{2}= 4 and

*y*

^{3}= –8

Column A | Column B |

x |
y |

*Which column is greater?*

*A. Column A
B. Column B
C. Both are Equal
D. Cannot be Determined*

*y*^{3} = –8 yields one cube root, *y* = –2. However, *x*^{2} = 4 yields two square roots, *x* = ±2. Now, if *x* = 2, then Column A is larger; but if *x* = –2, then the columns are equal.

This is a double case and the answer is (D).

*If x < 0 and y is 5 more than the square of x, which one of the following expresses x in terms of y?*

*A. *

B.

C.

D.

E.

Translating the expression *“y is 5 more than the square of x”* into an equation yields:

*y* = *x*^{2} + 5

*y* – 5 = *x*^{2}

Since we are given that *x* < 0, we take the negative root, . The answer is (B).