# Chapter 13 - Radicals and Radical Equations

This chapter discusses radicals. It contains the following sections:- section 13.1 - In this section we introduce radicals and talk about how to simplify them and do operations with them.
- section 13.2 - In this section we explain how to solve radical equations.

## 13.1 - Simplifying Radicals

You may want to review the sections on integer exponents and exponents in general before you read this section.### Radical notation

Here is a picture of a radical defining its parts:**Note:**

- The index,
*n*, must be a positive integer. - The above radical is spoken as “the
*n*^{th}root of*b*”, for any index except*n*= 2 when we say “the square root of*b*” and*n*= 3 when we say “the cube root of*b*”. - An index of 2, for the square root, is usually not written.
- The words “radical” and “root” will be used interchangably in this section.
- The roots in this section have (almost) nothing to do with roots of an equation.

A radical is merely an alternative way to write an
exponential whose exponent is a reciprocal .
It is defined like this:
This means that for every property or rule that holds for an exponential
there is a corresponding property or rule for a radical. |

### Root of a power and power of a root

A base raised to a fractional exponent*m*/

*n*can be written in two ways in terms of radicals:

- as the
*n*^{th}root of the*m*^{th}power: - or as the
*m*^{th}power of the*n*^{th}root:

*n*

^{th}power of the

*n*

^{th}root of any quantity and the

*n*

^{th}root of the

*n*

^{th}power of any quantity just equal that quantity:

and .Thus the

*n*

^{th}power and the

*n*

^{th}root are inverse functions.

### Root of a product or quotient

In section 3.3 we saw that a power of a product could be rewritten as a product of powers and a power of a quotient could be rewritten as a quotient of powers. For example: andThis means that the root of a product can be rewritten as a product of roots: and the root of a quotient can be rewritten as a quotient of roots: |

**Examples:**The properties of radicals given above can be used to simplify the expressions on the left to give the expressions on the right.

### Simplest form of a radical

A radical is said to be in simplest form (or standard form) when:- The radicand has been reduced as much as possible. (See the first example above.)
This is done by removing factors from the radical.

- There are no radicals in the denominator and no fractional radicands.
This is done by rationalizing the denominator.

- There are no products of radicals.
(See the second example above.)

- The index has been made as small as possible.

### Removing factors from the radicand

Suppose that the index of the radical is*n*. Then factor the radicand so that one or more of the factors is a perfect

*n*

^{th}power. Then rewrite the root of the product as a product of roots and use the fact that to simplify those factors. This process is called

**removing factors from the radicand**.

**Examples:**All of the following are

*square*roots. Therefore we look for perfect

*square*factors in the radicand. In the first example the factor 25 is a perfect square. In the second example the factor

*x*

^{ 4}is a perfect square. In the third example we factor out 4 rather than 20 because 4 is a perfect square whereas 20 is not. In the last example the entire radicand is a perfect square.

### Rationalizing the denominator

An expression is considered to be simpler when its*denominator*contains no radicals.

Suppose that the denominator of a fraction contains a

*square*root. Then multiply both the numerator and denominator of the fraction by that square root and simplify. This may produce a radical in the numerator but it will eliminate the radical from the denominator. This process is called

**rationalizing the denominator**.

**Example:**Simplify the expression .

**Solution:**Here are the steps:

- Multiply the numerator and denominator by the square root of 5 (shown below in blue).
- Multiply the fractions.
- Use the fact that . The result is a radical in the numerator but none in the denominator.

Suppose that a square root contains a fraction. Then multiply both the numerator and denominator of the fraction by the denominator of the fraction and simplify. This may produce a radical in the numerator but it will eliminate the radical from the denominator. This process is also called

**rationalizing the denominator**.

**Example:**Simplify the expression .

**Solution:**Here are the steps:

- The square root contains a fraction and the denominator of the fraction is 5
*y*. Multiply the numerator and denominator*of the fraction*by 5*y*(shown below in blue). - Multiply the fractions.
- Convert the root of a quotient to a quotient of roots.
- Use the fact that . The result is a radical remaining in the numerator but none in the denominator.

### Combining products of radicals

We saw above that the root of a product could be rewritten as a product of roots. Here we want to go the other way. It sometimes happens that converting a product of roots to the root of a product produces a perfect square factor and that factor can then be removed from the radicand.**Example:**Simplify the expression .

**Solution:**Write this as the root of a product, namely of 60, and then notice that 60 has a perfect square factor of 4, which can then be removed from the radicand.

### Reducing the index

It is sometimes possible to reduce the index by writing the radical in exponential form and then reducing the fractional exponent to lowest terms.**Example:**Simplify the expression .

**Solution:**Write this in exponential notation and use the exponentiation property of exponents to give the single exponent 2/6 and then reduce the fraction 2/6 to 1/3.

### Binomial denominators containing radicals

Suppose that the denominator of a fraction is a binomial (i.e. it contains two terms) and that one or both of those terms is a radical. Then multiplying the numerator and denominator of the fraction by the**binomial conjugate**of the denominator and distributing will eliminate all radicals from the denominator. (Note that the binomial conjugate of the binomial

*a*+

*b*is the binomial

*a*−

*b*and

*vice versa*.

**Example:**Eliminate the radical from the denominator of the expression .

**Solution:**Follow these steps:

- Multiply the numerator and denominator by the binomial conjugate of the denominator (shown below in blue).
- Multiply the fractions.
- Distribute in the numerator. In general the radicals survive. Distribute in the denominator. The cross-terms cancel so that the radicals disappear, which is the whole point of using the binomial conjugate.

## 13.2 - Radical Equations

Before reading this section you may want to review the following topics: A**radical equation**is one in which the unknown, call it

*x*, is inside a radical. To solve a radical equation follow these steps:

- Isolate the term containing the radical on one side of the equation,
say on the left-hand-side. (By isolate we mean get it by itself.)

- If the radical is a square root then square both sides of the equation.
(In general, if the radical is an
*n*^{th}root then take the*n*^{th}power of both sides.) On the left-hand-side use the fact that to get rid of the radical. On the right-hand-side distribute.

- The unknown
*x*is no longer inside a radical. Now you can finish solving for*x*by using the basic procedures for solving equations.

**Note:**

- If the equation contains more than one radical term, you will have to
perform the above procedure several times.

- One of the steps, squaring both sides of the equation, increases the degree of
the unknown, and this often leads to extraneous solutions. Therefore it is
**very important to check your solutions**.

**Example:**Solve the radical equation .

**Solution:**The first step is to isolate the radical term: Then square both sides:

6The result is that this is no longer a radical equation; it is a quadratic equation. Divide both sides by 2 to simplify it and then collect all terms on one side of the equation to put it into standard form:x+ 4 = 4x^{ 2}.

2Click here to see how the left-hand-side of this quadratic equation can be factored. The result is this:x^{ 2}− 3x− 2 = 0.

(The purpose of factoring is to put the equation into the formx− 2) (2x+ 1) = 0.

*a*·

*b*= 0. We can now replace it with two new equations. Each new equation comes from setting one of the factors to zero:

Their solutions arex− 2 = 0

2x+ 1 = 0

*x*= 2 and

*x*= −½.

**We must check these solutions.**Substituting

*x*= 2 into the original equation and simplifying causes the equation to read 0 = 0, so this solution checks out. But substituting

*x*= −½ back into the original equation causes it to read 2 = 0, so this solution doesn’t check out and must be rejected. Thus this radical equation has the single solution

*x*= 2.

**Example:**Solve the radical equation .

**Solution:**The first step is to isolate one of the radical terms (it doesn’t matter which one): Then square both sides: On the left-hand-side squaring got rid of the radical. On the right-hand-side we will have to distribute. After distributing and collecting like terms we get this equation: This is still a radical equation because although one radical is gone, the other one still remains. So we repeat the entire process. Isolate the remaining radical term (by canceling

*x*terms and dividing by 32): Then square both sides:

Note that this is no longer a radical equation; both radicals are now gone. Solving this equation yields the solutionx− 32 = 49

*x*= 81.

**We must now check the solution.**Substituting

*x*= 81 back into the original equation and simplifying gives the equation 16 = 16, so this solution checks out.