# 2.5: Square Roots and Real Numbers

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

## Learning Objectives

- Find square roots.
- Approximate square roots.
- Identify irrational numbers.
- Classify real numbers.
- Graph and order real numbers.

## Find Square Roots

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 1**

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

a) 121

b) 225

c) 324

d) 576

**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*}.

d) \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*}.

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 2**

Find the principal square root of the following numbers.

a) 8

b) 48

c) 75

d) 216

**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*}.

d) \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*}.

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 3**

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 4**

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*}

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.)

## Identify Irrational Numbers

Not all square roots are irrational, but any square root that can’t be reduced to a form with no radical signs in it is irrational. For example, \begin{align*}\sqrt{49}\end{align*} is rational because it equals 7, but \begin{align*}\sqrt{50}\end{align*} can’t be reduced farther than \begin{align*}5 \sqrt{2}\end{align*}. That factor of \begin{align*}\sqrt{2}\end{align*} is irrational, making the whole expression irrational.

**Example 5**

*Identify which of the following are rational numbers and which are irrational numbers.*

a) 23.7

b) 2.8956

c) \begin{align*}\pi\end{align*}

d) \begin{align*}\sqrt{6}\end{align*}

e) \begin{align*}3. \overline{27}\end{align*}

**Solution**

a) 23.7 can be written as \begin{align*}23 \frac{7}{10}\end{align*}, so it is rational.

b) 2.8956 can be written as \begin{align*}2 \frac{8956}{10000}\end{align*}, so it is rational.

c) \begin{align*}\pi = 3.141592654 \ldots\end{align*} We know from the definition of \begin{align*}\pi\end{align*} that the decimals do not terminate or repeat, so \begin{align*}\pi\end{align*} is irrational.

d) \begin{align*}\sqrt{6} = \sqrt{2} \ \times \sqrt{3}\end{align*}. We can’t reduce it to a form without radicals in it, so it is irrational.

e) \begin{align*}3. \overline{27} = 3.272727272727 \ldots\end{align*} This decimal goes on forever, but it’s not random; it repeats in a predictable pattern. Repeating decimals are always rational; this one can actually be expressed as \begin{align*}\frac{36}{11}\end{align*}.

You can see from this example that any number whose decimal representation has a finite number of digits is rational, since each decimal place can be expressed as a fraction. For example, 0.439 can be expressed as \begin{align*}\frac{4}{10} + \frac{3}{100} + \frac{9}{1000}\end{align*}, or just \begin{align*}\frac{439}{1000}\end{align*}. Also, any decimal that repeats is rational, and can be expressed as a fraction. For example, \begin{align*}0.25 \overline{38}\end{align*} can be expressed as \begin{align*}\frac{25}{100} + \frac{38}{9900}\end{align*}, which is equivalent to \begin{align*}\frac{2513}{9900}\end{align*}.

## Classify Real Numbers

We can now see how real numbers fall into one of several categories.

If a real number can be expressed as a rational number, it falls into one of two categories. If the denominator of its **simplest form** is one, then it is an **integer**. If not, it is a fraction (this term also includes decimals, since they can be written as fractions.)

If the number cannot be expressed as the ratio of two integers (i.e. as a fraction), it is **irrational**.

**Example 6**

*Classify the following real numbers.*

a) 0

b) -1

c) \begin{align*}\frac{\pi}{3}\end{align*}

d) \begin{align*}\frac{\sqrt{2}}{3}\end{align*}

e) \begin{align*}\frac{\sqrt{36}}{9}\end{align*}

**Solution**

a) Integer

b) Integer

c) Irrational (Although it’s written as a fraction, \begin{align*}\pi\end{align*} is irrational, so any fraction with \begin{align*}\pi\end{align*} in it is also irrational.)

d) Irrational

e) Rational (It simplifies to \begin{align*}\frac{6}{9}\end{align*}, or \begin{align*}\frac{2}{3}\end{align*}.)

## Lesson Summary

- 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.

## Review Questions

- 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*}

- 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*}

- Classify the following numbers as an integer, a rational number or an irrational number.
- \begin{align*}\sqrt{0.25}\end{align*}
- \begin{align*}\sqrt{1.35}\end{align*}
- \begin{align*}\sqrt{20}\end{align*}
- \begin{align*}\sqrt{25}\end{align*}
- \begin{align*}\sqrt{100}\end{align*}

- Place the following numbers in numerical order, from lowest to highest. \begin{align*}\frac{\sqrt{6}}{2} \qquad \frac{61}{50} \qquad \sqrt{1.5} \qquad \frac{16}{13}\end{align*}
- Use the marked points on the number line and identify each proper fraction.