### Square Roots and Irrational Numbers

The **square root** of a number is a number which, when multiplied by itself, gives the original number. In other words, if , we say that is the square root of .

**Note:** Negative numbers and positive numbers both yield positive numbers when squared, so each positive number has both a positive and a negative square root. (For example, 3 and -3 can both be squared to yield 9.) The positive square root of a number is called the **principal square root**.

The square root of a number is written as or sometimes as . The symbol is sometimes called a **radical sign**.

Numbers with whole-number square roots are called **perfect squares**. The first five perfect squares (1, 4, 9, 16, and 25) are shown below.

You can determine whether a number is a perfect square by looking at its prime factors. If every number in the factor tree appears an even number of times, the number is a perfect square. To find the square root of that number, simply take one of each pair of matching factors and multiply them together.

#### Finding the Principal Square Root

Find the principal square root of each of these perfect squares.

a) 121

, so .

b) 225

, so .

c) 324

, so .

When the prime factors don’t pair up neatly, we “factor out” the ones that do pair up and leave the rest under a radical sign. We write the answer as , where is the product of half the paired factors we pulled out and is the product of the leftover factors.

#### Finding the Principal Square Roots of Imperfect Squares

Find the principal square root of the following numbers.

a) 8

. This gives us one pairs of 2's and one leftover 2, so .

b) 48

, so , or .

c) 75

, so .

Note that in the last example we collected the paired factors first, **then** we collected the unpaired ones under a single radical symbol. Here are the four rules that govern how we treat square roots.

#### Simplifying Square Roots

Simplify the following square root problems

a)

b)

c)

d)

**Approximate Square Roots**

Terms like and (square roots of prime numbers) cannot be written as **rational numbers**. That is to say, they cannot be expressed as the ratio of two integers. We call them **irrational numbers**. In decimal form, they have an unending, seemingly random, string of numbers after the decimal point.

To find approximate values for square roots, we use the or button on a calculator. When the number we plug in is a perfect square, or the square of a rational number, we will get an exact answer. When the number is a non-perfect square, the answer will be irrational and will look like a random string of digits. Since the calculator can only show some of the infinitely many digits that are actually in the answer, it is really showing us an **approximate answer**—not exactly the right answer, but as close as it can get.

#### Using a Calculator

Use a calculator to find the following square roots. Round your answer to three decimal places.

a)

b)

c)

d)

### Examples

Find the square root of each number.

#### Example 1

576

, so

#### Example 2

216

, so , or .

### Review

For 1-10, find the following square roots **exactly without using a calculator**, giving your answer in the simplest form.

- (Hint: The division rules you learned can be applied backwards!)

For 11-20, use a calculator to find the following square roots. Round to two decimal places.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 2.9.