Introduction

A radical, or “square root,” is the opposite of an exponent.  If 2² = 4, then = 2.  The symbol is called a “square root” andis called “the square root of four.”  The following chart shows some exponents as well their opposite radical expressions.

Each of the above exponential expressions was taken to the second power.  Exponents that are taken to the 3^rd, 4^th, 5^th, or larger powers can be undone by taking 3^rd, 4^th, 5^th, and larger roots as shown in the chart below.

The majority of the radicals you will see in this lesson are to the 2^nd power and are called square roots.  To find the value of a square root, identify a (smaller) number that can be multiplied by itself to yield the (larger) result inside the radical.

Lesson

A radical expression is simply an expression that contains a square root sign.  Many radical expressions contain a single number inside the square root sign and can be simplified down to a single integer.

A number is called a perfect square if its square root is a whole number.  The following numbers are the perfect squares whose square roots are 15 or less.

Square Root of a Product

When a number is a perfect square, its square root can be simplified down to a single whole number.  Square roots of numbers that are not perfect squares can often be simplified, but the result is not a whole number.  Before doing an example, take a look at the following mathematical property of radicals:

This property is true as long as a and b are positive.  This property can be used to simplify the following problems:

            -simplify

             -simplify 

The first answer can be expressed in words as “two square roots of three” and the second as “five square roots of two.”  The best way to do the problems is to think of potential perfect square factors of the number in the radical.  In the first example, 12 = 4 × 3 and 12 = 6 × 2.  In the problem, use the factors 4 × 3 since 4 is a perfect square.

Square root of a Quotient

The square root of a fraction (or quotient) can be simplified by taking the square root of the numerator and denominator separately.  This fact is expressed by the following property:

This property is true as long as a and b are positive.  The square root of a quotient property can be used to simplify the following problems:

            -simplify

            -simplify

Review

Simplify each radical expression:

1) 

2) 

3) 

4)   

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Answers:

1) 

2) 

3) 

4)  

*Note that you may not have a radical in the denominator of your answer.  You will learn about rationalizing the denominator in a future lesson.