# At a Glance - Addition and Subtraction

To add or subtract radical expressions, simplify each radical term and then combine like terms. A simplified radical term consists of a coefficient and a radical, under which there is a radicand. Can you believe you understand now what all these crazy words mean? It's like you can speak a secret language.

### Sample Problem

In the term , the coefficient is 5 and the radicand is 7.

### Sample Problem

In the term , we need to rationalize the denominator to find .

To put this in the form we need, it could be rewritten as:

So the coefficient is and the radicand is 35. We still haven't gotten rid of the fraction line, but at least it isn't combined with the square root symbol any longer. That many weird symbols consorting together makes us nervous. It feels like they're up to something.

Two radical terms are considered **like terms** if they have the same radicand. This makes them "term twins." You'll be able to tell, because they're always finishing each other's sentences.

**Be careful:** It's important to simplify radical terms *before* combining like terms. Sometimes two terms can be rewritten as like terms, but we can't see it until we simplify. It's the same way you can't switch one kid out for another without their parents noticing until you've first made sure that they're twins. Otherwise, it's called kidnapping.

Radical expressions may contain variables either outside or inside the radicands. After simplifying, we can combine like terms in the same way we did when only numbers were involved. Except now it's more fun, because we can use variables!

...we'll keep telling ourselves that until it feels true.

Now that we know how to figure out which terms can be combined, we'll combine some. How about that.

This is similar to adding or subtracting variables. In the same way that 3*x* + 4*x* = 7*x*, so does .

To add or subtract like radical terms, we add or subtract the coefficients. We don't do anything to the radicands, which is why we made sure they were the same in the first place.

### Sample Problem

Add .

We keep the radicand the same, and add the coefficients 3 and 8:

=

### Sample Problem

What's ?

First, simplify each radical. Then we can rewrite the problem as:

=

=

### Sample Problem

What's ?

We can simplify the first radical term and rewrite the problem as:

We can't combine these terms since the radicands aren't the same, so that's our final answer.

We can also do this sort of thing with expressions that have variables. Variables and numbers have sort of an "anything you can do, I can do better" relationship, or at least an "anything you can do, I can do equally" one.

#### Example 1

Can the following be written as like terms? and |

#### Example 2

Can the following be written as like terms? and |

#### Example 3

Can the following be written as like terms? and |

#### Example 4

Can the following be written as like terms? and |

#### Example 5

Can the following be written as like terms? and |

#### Example 6

Can the following be written as like terms? and |

#### Example 7

Can the following be written as like terms? and |

#### Example 8

Can the following be written as like terms? and |

#### Example 9

Can the following be written as like terms? and |

#### Example 10

Can the following be written as like terms? and |

#### Example 11

Do the arithmetic and simplify your answer. |

#### Example 12

Do the arithmetic and simplify your answer. |

#### Example 13

Do the arithmetic and simplify your answer. |

#### Example 14

Do the arithmetic and simplify your answer. |

#### Exercise 1

Simplify the radical term. What is the coefficient and radicand of the simplified form?

#### Exercise 2

Simplify the radical term. What is the coefficient and radicand of the simplified form?

#### Exercise 3

Simplify the radical term. What is the coefficient and radicand of the simplified form?

#### Exercise 4

Simplify the radical term. What is the coefficient and radicand of the simplified form?

#### Exercise 5

Simplify the radical term. What is the coefficient and radicand of the simplified form?

#### Exercise 6

Can the following be written as like terms?

and

#### Exercise 7

Can the following be written as like terms?

and

#### Exercise 8

Can the following be written as like terms?

and

#### Exercise 9

Can the following be written as like terms?

and

#### Exercise 10

Can the following be written as like terms?

and

#### Exercise 11

Can the following be written as like terms?

and

#### Exercise 12

Can the following be written as like terms?

and

#### Exercise 13

Can the following be written as like terms?

and

#### Exercise 14

Can the following be written as like terms?

and

#### Exercise 15

Can the following be written as like terms?

and

#### Exercise 16

Do the arithmetic and simplify your answer: .

#### Exercise 17

Do the arithmetic and simplify your answer: .

#### Exercise 18

Do the arithmetic and simplify your answer: .

#### Exercise 19

Do the arithmetic and simplify your answer: .

#### Exercise 20

Do the arithmetic and simplify your answer: .

#### Exercise 21

Do the arithmetic and simplify your answer: .

#### Exercise 22

Do the arithmetic and simplify your answer: .

#### Exercise 23

Do the arithmetic and simplify your answer: .

#### Exercise 24

Do the arithmetic and simplify your answer: .

#### Exercise 25

Do the arithmetic and simplify your answer: .