# Roots

The symbol is read the nth 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 nth power yields b. In other words, a is the nth root of b if .
For example,
* because , and  because .

Even roots occur in pairs: both a positive root and a negative root.
For example,
since , and  since .

Odd roots occur alone and have the same sign as the base:  since . 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

Square roots and cube roots can be simplified by removing perfect squares and perfect cubes, respectively.
For example,

Radicals are often written with fractional exponents. The expression  can be written as . This can be generalized as follows:
Usually, the form  is better when calculating because the part under the radical is smaller in this case.
For example, .

Using the form  would be much harder in this case: . Most students know the value of , but few know the value of If n is even, then

For example, . With odd roots, the absolute value symbol is not needed.
For example,
.

To solve radical equations, just apply the rules of exponents to undo the radicals.
For example, to solve the radical equation
, we cube both sides to eliminate the cube root:

Even roots of negative numbers do not appear on the SAT.
For example, you will not see expressions of the form ; expressions of this type are called complex numbers.

The following rules are useful for manipulating roots:

 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.

To add two roots, both the index and the base must be the same.
For example,  cannot be added because the indices are different, nor can  be added because the bases are different. However, .

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.
For example, .

You need to know the approximations of the following roots:

Example

Given the system, which of the following is NOT necessarily true?

1. y < 0
2. x < 5
3. y is an integer
4. x > y
5. x/y is an integer
Solution

y3 = –8 yields one cube root, y = –2.
However, x2 = 4 yields two square roots, x = ±2.
Now, if x = 2, then x > y; but if x = –2, then x = y.
Hence, choice (D) is not necessarily true.

Example

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

Solution

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

y = x2 + 5
y – 5 = x2

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