2.12: 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? In this Concept, you'll learn how to take the square root of a number and decide whether the result is rational or irrational so that you can answer questions such as these.
Guidance
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?
To answer this question, you will need to know how to find the square root of a number.
The square root of a number \begin{align*}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*}
For example:
\begin{align*}\sqrt{81}=9\end{align*}
\begin{align*}\sqrt{81}= 9\end{align*}
\begin{align*}\pm \sqrt{81} = \pm 9\end{align*}
Example A
The human chessboard measures 324 square meters. How long is one side of the square?
Solution: The area of a square is \begin{align*}s^2 = Area\end{align*}
\begin{align*}s^2=324\end{align*}
The value of \begin{align*}s\end{align*}
\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*}
There is no whole number multiplied by itself that equals five, so \begin{align*}\sqrt{5}\end{align*}
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.
Example B
Estimate \begin{align*}\sqrt{5}\end{align*}
Solution: The perfect square below 5 is 4 and the perfect square above 5 is 9. Therefore, \begin{align*}4<5<9\end{align*}
Identifying Irrational Numbers
Recall the number hierarchy from a previous Concept. Real numbers have two categories: rational and irrational. If a value 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 (nonterminating) and does not repeat a pattern (nonrepeating). 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*}
Begin by writing the prime factorization of \begin{align*}\sqrt{18}\end{align*}
You can check your answer on a calculator by finding the decimal approximation for each square root.
Example C
Find the exact value of \begin{align*}\sqrt{75}\end{align*}
Solution:
\begin{align*}\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5 \sqrt{3}\end{align*}
Video Review
Guided Practice
The area of a square is 50 square feet. What are the lengths of its sides?
Solution:
We know that the formula for the area of a square is \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*}
Practice
Find the following square roots exactly without using a calculator. Give your answer in the simplest form.

\begin{align*}\sqrt{25}\end{align*}
25−−√ 
\begin{align*}\sqrt{24}\end{align*}
24−−√ 
\begin{align*}\sqrt{20}\end{align*}
20−−√ 
\begin{align*}\sqrt{200}\end{align*}
200−−−√ 
\begin{align*}\sqrt{2000}\end{align*}
2000−−−−√  \begin{align*}\sqrt{\frac{1}{4}}\end{align*}
 \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{0.25}\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*}
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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
Here you'll learn how to decide whether a number is rational or irrational and how to take the square root of a number.