<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Square Roots and Irrational Numbers

Simplify square roots by factoring

Atoms Practice
Estimated15 minsto complete
%
Progress
Practice Square Roots and Irrational Numbers
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated15 minsto complete
%
Practice Now
Turn In
Square Roots and Irrational Numbers

Suppose an elementary school has a square playground with an area of 3000 square feet. Could you find the width of the playground? Would the width be a rational or irrational number

Square Roots and Irrational Numbers

The square root of a number \begin{align*}n\end{align*} is any number \begin{align*}s\end{align*} such that \begin{align*}s^2 = n\end{align*}.

Every positive number has two square roots, the positive and the negative. The symbol used to represent the square root is \begin{align*}\sqrt{x}\end{align*}. It is assumed that this is the positive square root of \begin{align*}x\end{align*}. To show both the positive and negative values, you can use the symbol \begin{align*}\pm\end{align*}, read “plus or minus.”

For example:

\begin{align*}\sqrt{81}=9\end{align*} means the positive square root of 81.

\begin{align*}-\sqrt{81}= -9\end{align*} means the negative square root of 81.

\begin{align*}\pm \sqrt{81} = \pm 9\end{align*} means the positive or negative square root of 81.

Let's solve the following problem by using square roots:

Human chess is a variation of chess, often played at Renaissance fairs, in which people take on the roles of the various pieces on a chessboard. The chessboard is played on a square plot of land that measures 324 square meters with the chess squares marked on the grass. How long is each side of the chessboard? 

The human chessboard measures 324 square meters. 

The area of a square is \begin{align*}s^2 = Area\end{align*}. The value of Area can be replaced with 324.

\begin{align*}s^2=324\end{align*}

The value of \begin{align*}s\end{align*} represents the square root of 324.

\begin{align*}s= \sqrt{324}=18\end{align*}

The chessboard is 18 meters long by 18 meters wide.

Approximating Square Roots

When the square root of a number is a whole number, this number is called a perfect square. 9 is a perfect square because \begin{align*}\sqrt{9}=3\end{align*}.

Not all square roots are whole numbers. Many square roots are irrational numbers, meaning there is no rational number equivalent. For example, 2 is the square root of 4 because \begin{align*}2 \times 2 = 4\end{align*}. The number 7 is the square root of 49 because \begin{align*}7 \times 7 = 49\end{align*}. What is the square root of 5?

There is no whole number multiplied by itself that equals five, so \begin{align*}\sqrt{5}\end{align*} is not a whole number. To find the value of \begin{align*}\sqrt{5}\end{align*}, we can use estimation.

To estimate the square root of a number, look for the perfect integers less than and greater than the value, and then estimate the decimal.

Let's estimate \begin{align*}\sqrt{5}\end{align*}:

The perfect square below 5 is 4 and the perfect square above 5 is 9. Therefore, \begin{align*}4<5<9\end{align*}. Therefore, \begin{align*}\sqrt{5}\end{align*} is between \begin{align*}\sqrt{4}\end{align*} and \begin{align*}\sqrt{9}\end{align*}, or \begin{align*}2< \sqrt{5}<3\end{align*}. Because 5 is closer to 4 than 9, the decimal is a low value: \begin{align*}\sqrt{5} \approx 2.2\end{align*}.

Identifying Irrational Numbers

Real numbers have two categories: rational and irrational. If a square root is not a perfect square, then it is considered an irrational number. These numbers cannot be written as a fraction because the decimal does not end (non-terminating) and does not repeat a pattern (non-repeating). Although irrational square roots cannot be written as fractions, we can still write them exactly, without typing the value into a calculator.

For example, suppose you do not have a calculator and you need to find \begin{align*}\sqrt{18}\end{align*}. You know there is no whole number squared that equals 18, so \begin{align*}\sqrt{18}\end{align*} is an irrational number. The value is between \begin{align*}\sqrt{16}=4\end{align*} and \begin{align*}\sqrt{25}=5\end{align*}. However, we need to find the exact value of \begin{align*}\sqrt{18}\end{align*}.

Begin by writing the prime factorization of \begin{align*}\sqrt{18}\end{align*}. \begin{align*}\sqrt{18} = \sqrt{9 \times 2}= \sqrt{9} \times \sqrt{2}\end{align*}. \begin{align*}\sqrt{9} = 3\end{align*} but \begin{align*}\sqrt{2}\end{align*} does not have a whole number value. Therefore, the exact value of \begin{align*}\sqrt{18}\end{align*} is \begin{align*}3 \sqrt{2}\end{align*}.

You can check your answer on a calculator by finding the decimal approximation for each square root.

Let's find the exact value of \begin{align*}\sqrt{75}\end{align*}:

\begin{align*}\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5 \sqrt{3}\end{align*}

   

 

 

Examples

Example 1

Earlier, you were told that an elementary school has a square playground with an area of 3000 square feet. What is the width of the playground? Is the width a rational or irrational number?

Since the playground is a square, we can use the formula for the area of a square: \begin{align*}a=s^2.\end{align*}

\begin{align*}& a=s^2\\ &3000=s^2\\ &\sqrt{3000}=\sqrt{s^2}\\ &\sqrt{3000}=s \end{align*}

Now, simplify:

\begin{align*}&\sqrt{100\times 30}=s\\ &\sqrt{100}\times\sqrt{ 30}=s\\ &10\sqrt{30}=s\end{align*} 

The width of the playground is \begin{align*}10\sqrt{30}\end{align*} feet. Since 30 is not a perfect square, the width is irrational.

Example 2

The area of a square is 50 square feet. What are the lengths of its sides?

 Using the area of a square formula, \begin{align*}a=s^2\end{align*}:

\begin{align*}& a=s^2\\ &50=s^2\\ &\sqrt{50}=\sqrt{s^2}\\ &\sqrt{50}=s\end{align*}

Now we will simplify:

\begin{align*}\sqrt{50}=\sqrt{25\cdot 2}=5\sqrt{2}.\end{align*}

The length of each side of the square is \begin{align*}5\sqrt{2}\end{align*} feet.

Review

Find the following square roots exactly without using a calculator. Give your answer in the simplest form.

  1. \begin{align*}\sqrt{25}\end{align*}
  2. \begin{align*}\sqrt{24}\end{align*}
  3. \begin{align*}\sqrt{20}\end{align*}
  4. \begin{align*}\sqrt{200}\end{align*}
  5. \begin{align*}\sqrt{2000}\end{align*}
  6. \begin{align*}\sqrt{\frac{1}{4}}\end{align*}
  7. \begin{align*}\sqrt{\frac{9}{4}}\end{align*}
  8. \begin{align*}\sqrt{0.16}\end{align*}
  9. \begin{align*}\sqrt{0.1}\end{align*}
  10. \begin{align*}\sqrt{0.01}\end{align*}

Use a calculator to find the following square roots. Round to two decimal places.

  1. \begin{align*}\sqrt{13}\end{align*}
  2. \begin{align*}\sqrt{99}\end{align*}
  3. \begin{align*}\sqrt{123}\end{align*}
  4. \begin{align*}\sqrt{2}\end{align*}
  5. \begin{align*}\sqrt{2000}\end{align*}
  6. \begin{align*}\sqrt{0.25}\end{align*}
  7. \begin{align*}\sqrt{1.35}\end{align*}
  8. \begin{align*}\sqrt{0.37}\end{align*}
  9. \begin{align*}\sqrt{0.7}\end{align*}
  10. \begin{align*}\sqrt{0.01}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.12. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Vocabulary

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.

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.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Square Roots and Irrational Numbers.
Please wait...
Please wait...