# 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 \begin{align*}a = b^2\end{align*}, we say that \begin{align*}b\end{align*} is the square root of \begin{align*}a\end{align*}.

**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 \begin{align*}x\end{align*} is written as \begin{align*}\sqrt{x}\end{align*} or sometimes as \begin{align*}\sqrt[2]{x}\end{align*}. The symbol \begin{align*}\sqrt{\;\;}\end{align*} 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) \begin{align*}121 = 11 \times 11\end{align*}, so \begin{align*}\sqrt{121} = 11\end{align*}.

b) \begin{align*}225 = (5 \times 5) \times (3 \times 3)\end{align*}, so \begin{align*}\sqrt{225} = 5 \times 3 = 15\end{align*}.

c) \begin{align*}324 = (2 \times 2) \times (3 \times 3) \times (3 \times 3)\end{align*}, so \begin{align*}\sqrt{324} = 2 \times 3 \times 3 = 18\end{align*}.

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 \begin{align*}a \sqrt{b}\end{align*}, where \begin{align*}a\end{align*} is the product of half the paired factors we pulled out and \begin{align*}b\end{align*} 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) \begin{align*}8 = 2 \times 2 \times 2\end{align*}. This gives us one pair of 2’s and one leftover 2, so \begin{align*}\sqrt{8} = 2 \sqrt{2}\end{align*}.

b) \begin{align*}48 = (2 \times 2) \times (2 \times 2) \times 3\end{align*}, so \begin{align*}\sqrt{48} = 2 \times 2 \times \sqrt{3}\end{align*}, or \begin{align*}4 \sqrt{3}\end{align*}.

c) \begin{align*}75 = (5 \times 5) \times 3\end{align*}, so \begin{align*}\sqrt{75} = 5 \sqrt{3}\end{align*}.

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.

- \begin{align*}\sqrt{a} \times \sqrt{b} = \sqrt{ab}\end{align*}
- \begin{align*}A \sqrt{a} \times B \sqrt{b} = AB \sqrt{ab}\end{align*}
- \begin{align*}\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\end{align*}
- \begin{align*}\frac{A \sqrt{a}}{B \sqrt{b}} = \frac{A}{B} \sqrt{\frac{a}{b}}\end{align*}

#### Example C

Simplify the following square root problems

a) \begin{align*}\sqrt{8} \times \sqrt{2}\end{align*}

b) \begin{align*}3 \sqrt{4} \times 4 \sqrt{3}\end{align*}

c) \begin{align*}\sqrt{12} \ \div \sqrt{3}\end{align*}

d) \begin{align*}12 \sqrt{10} \div 6 \sqrt{5}\end{align*}

**Solution**

a) \begin{align*}\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4\end{align*}

b) \begin{align*}3 \sqrt{4} \times 4 \sqrt{3} = 12 \sqrt{12} = 12 \sqrt{(2 \times 2) \times 3} = 12 \times 2 \sqrt{3} = 24 \sqrt{3}\end{align*}

c) \begin{align*}\sqrt{12} \ \div \sqrt{3} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2\end{align*}

d) \begin{align*}12 \sqrt{10} \div 6 \sqrt{5} = \frac{12}{6} \sqrt{\frac{10}{5}} = 2 \sqrt{2}\end{align*}

**Approximate Square Roots**

Terms like \begin{align*}\sqrt{2}, \sqrt{3}\end{align*} and \begin{align*}\sqrt{7}\end{align*} (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 \begin{align*}\sqrt{\;\;}\end{align*} or \begin{align*}\sqrt{x}\end{align*} 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) \begin{align*}\sqrt{99}\end{align*}

b) \begin{align*}\sqrt{5}\end{align*}

c) \begin{align*}\sqrt{0.5}\end{align*}

d) \begin{align*}\sqrt{1.75}\end{align*}

**Solution**

a) \begin{align*}\approx 9.950\end{align*}

b) \begin{align*}\approx 2.236\end{align*}

c) \begin{align*}\approx 0.707\end{align*}

d) \begin{align*}\approx 1.323\end{align*}

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 \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, if \begin{align*}a = b^2\end{align*}, then \begin{align*}b = \sqrt{a}\end{align*}. - 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:
- \begin{align*}\sqrt{a} \ \times \sqrt{b} = \sqrt{ab}\end{align*}
- \begin{align*}A \sqrt{a} \ \times B \sqrt{b} = AB \sqrt{ab}\end{align*}
- \begin{align*}\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{ \frac{a}{b}}\end{align*}
- \begin{align*}\frac{A\sqrt{a}}{B \sqrt{b}} = \frac{A}{B} \sqrt{\frac{a}{b}}\end{align*}

- 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) \begin{align*}576 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3)\end{align*}, so \begin{align*}\sqrt{576} = 2 \times 2 \times 2 \times 3 = 24\end{align*}.

b) \begin{align*}216 = (2 \times 2) \times 2 \times (3 \times 3) \times 3\end{align*}, so \begin{align*}\sqrt{216} = 2 \times 3 \times \sqrt{2 \times 3}\end{align*}, or \begin{align*} 6 \sqrt{6}\end{align*}.

### Practice

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

- \begin{align*}\sqrt{25}\end{align*}
- \begin{align*}\sqrt{24}\end{align*}
- \begin{align*}\sqrt{20}\end{align*}
- \begin{align*}\sqrt{200}\end{align*}
- \begin{align*}\sqrt{2000}\end{align*}
- \begin{align*}\sqrt{\frac{1}{4}}\end{align*} (Hint: The division rules you learned can be applied backwards!)
- \begin{align*}\sqrt{\frac{9}{4}}\end{align*}
- \begin{align*}\sqrt{0.16}\end{align*}
- \begin{align*}\sqrt{0.1}\end{align*}
- \begin{align*}\sqrt{0.01}\end{align*}

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

- \begin{align*}\sqrt{13}\end{align*}
- \begin{align*}\sqrt{99}\end{align*}
- \begin{align*}\sqrt{123}\end{align*}
- \begin{align*}\sqrt{2}\end{align*}
- \begin{align*}\sqrt{2000}\end{align*}
- \begin{align*}\sqrt{.25}\end{align*}
- \begin{align*}\sqrt{1.35}\end{align*}
- \begin{align*}\sqrt{0.37}\end{align*}
- \begin{align*}\sqrt{0.7}\end{align*}
- \begin{align*}\sqrt{0.01}\end{align*}

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.