# 2.9: Square Roots and Irrational Numbers

**At Grade**Created by: CK-12

**Practice**Square Roots and Irrational Numbers

What if you had a number like 1000 and you wanted to find its square root? After completing this concept, you'll be able to find square roots like this one by hand and with a calculator.

### Watch This

CK-12 Foundation: 0209S Square Roots (H264)

### Try This

You can also work out square roots by hand using a method similar to long division. (See the web page at http://www.homeschoolmath.net/teaching/square-root-algorithm.php for an explanation of this method.)

### Guidance

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.

#### Example A

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

a) 121

b) 225

c) 324

**Solution**

a) , so .

b) , so .

c) , so .

For more practice matching numbers with their square roots, try the Flash games at http://www.quia.com/jg/65631.html.

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.

#### Example B

Find the principal square root of the following numbers.

a) 8

b) 48

c) 75

**Solution**

a) . This gives us one pair of 2’s and one leftover 2, so .

b) , so , or .

c) , 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.

#### Example C

Simplify the following square root problems

a)

b)

c)

d)

**Solution**

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.

#### Example D

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

a)

b)

c)

d)

**Solution**

a)

b)

c)

d)

Watch this video for help with the Examples above.

CK-12 Foundation: Square Roots

### Vocabulary

- The
**square root**of a number is a number which gives the original number when multiplied by itself. In algebraic terms, for two numbers and , if , then . - A square root can have two possible values: a positive value called the
**principal square root**, and a negative value (the opposite of the positive value). - A
**perfect square**is a number whose square root is an integer. - Some mathematical properties of square roots are:
- Square roots of numbers that are not perfect squares (or ratios of perfect squares) are
**irrational numbers**. They cannot be written as rational numbers (the ratio of two integers). In decimal form, they have an unending, seemingly random, string of numbers after the decimal point. - Computing a square root on a calculator will produce an
**approximate solution**since the calculator only shows a finite number of digits after the decimal point.

### Guided Practice

*Find the square root of each number.*

a) 576

b) 216

**Solution**

a) , so .

b) , so , or .

### Practice

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.

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approximate solution

An approximate solution to a problem is a solution that has been rounded to a limited number of digits.Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.Perfect Square

A perfect square is a number whose square root is an integer.principal square root

The principal square root is the positive square root of a number, to distinguish it from the negative value. 3 is the principal square root of 9; -3 is also a square root of 9, but it is not principal square root.rational number

A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.Square Root

The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.### Image Attributions

Here you'll learn how to find and approximate square roots. You'll also learn how to simplify expressions involving square roots.