Introduction

The square root of a number is the reverse of squaring a number. We express square roots using the radical symbol \sqrt{\phantom{x}}.

For example, since " {\color{blue}3} squared equals 9 "

{\color{blue}3}^2 = {\color{blue}3} \cdot {\color{blue}3} = 9,

then, "the square root of 9 equals {\color{blue}3} ," which we write as

\sqrt{9} = {\color{blue}3}.

We can think of the square root geometrically. Let's represent the number 9 by a collection of nine items:

We can pile up these items to form a square, where the length of each side equals \sqrt{9}={\color{blue}3}\mathbin{:}

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Perfect Squares Up to 144

Some other basic square roots are as follows: \begin{array}{} \sqrt{0} &=& \sqrt{{\color{blue}0}^2} &=& {\color{blue}0} \\ \sqrt{1} &=& \sqrt{{\color{blue}1}^2} &=& {\color{blue}1} \\ \sqrt{4} &=& \sqrt{{\color{blue}2}^2} &=& {\color{blue}2} \\ \sqrt{9} &=& \sqrt{{\color{blue}3}^2} &=& {\color{blue}3} \\ \sqrt{16} &=& \sqrt{{\color{blue}4}^2} &=& {\color{blue}4} \\ \sqrt{25} &=& \sqrt{{\color{blue}5}^2} &=& {\color{blue}5} \end{array}

A perfect square is a number whose square root is a whole number. So, the numbers 0, \qquad 1, \qquad 4, \qquad 9, \qquad 16,\qquad 25 are all perfect squares.

The perfect squares can be found on the diagonal entries of a multiplication table. For instance, the perfect squares up to 12^2 = 144 are highlighted below.


\times \mathbf 1 \mathbf 2 \mathbf 3 \mathbf 4 \mathbf 5 \mathbf 6 \mathbf 7 \mathbf 8 \mathbf 9 \mathbf{10} \mathbf{11} \mathbf{12}
\mathbf{1} 1 2 3 4 5 6 7 8 9 10 11 12
\mathbf{2} 2 4 6 8 10 12 14 16 18 20 22 24
\mathbf{3} 3 6 9 12 15 18 21 24 27 30 33 36
\mathbf{4} 4 8 12 16 20 24 28 32 36 40 44 48
\mathbf{5} 5 10 15 20 25 30 35 40 45 50 55 60
\mathbf{6} 6 12 18 24 30 36 42 48 54 60 66 72
\mathbf{7} 7 14 21 28 35 42 49 56 63 70 77 84
\mathbf{8} 8 16 24 32 40 48 56 64 72 80 88 96
\mathbf{9} 9 18 27 36 45 54 63 72 81 90 99 108
\mathbf{10} 10 20 30 40 50 60 70 80 90 100 110 120
\mathbf{11} 11 22 33 44 55 66 77 88 99 110 121 132
\mathbf{12} 12 24 36 48 60 72 84 96 108 120 132 144


Finally, if a represents a number that's either positive or zero (i.e., a\geq 0 ), then \sqrt{a^2} = a.

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Example: The Square Root of a Positive Number Squared

What is \sqrt{15^2}\,?

EXPLANATION

Recall that if a\geq 0, then

\sqrt{a^2} = a.

In our case, since the number under the square root ( \sqrt{\phantom{x}} ) is already written as a square, we obtain the answer immediately: \sqrt{15^2} = 15

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Practice: The Square Root of a Positive Number Squared

$\sqrt{14^2}=$

a
 $\pm 196$
b
 $\pm 14$
c
 $-14$
d
 $196$
e
 $14$
Practice: The Square Root of a Positive Number Squared

$\sqrt{\left(\dfrac{3}{5}\right)^2}=$

a
 $\dfrac{5}{3}$
b
 $\pm \dfrac{3}{5}$
c
 $\dfrac{1}{5}$
d
 $\dfrac{3}{5}$
e
 $\pm \dfrac{5}{3}$
Example: The Square Root of a Perfect Square

Calculate \sqrt{36}.

EXPLANATION

Recall that if a\geq 0, then

\sqrt{a^2} = a.

To work out \sqrt{36}, we express 36 as a perfect square:

\begin{align*} \sqrt{36} &=\\[5pt] \sqrt{6\cdot 6}&=\\[5pt] \sqrt{6^2} &=\\[5pt] 6& \end{align*}

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Practice: The Square Root of a Perfect Square

$\sqrt{4}=$

a
 $1$
b
 $4$
c
 $2$
d
 $16$
e
 $8$
Practice: The Square Root of a Perfect Square

$\sqrt{81}=$

a
 $81$
b
 $8$
c
 $-9$
d
 $\pm 9$
e
 $9$
Example: The Square Root of a Fraction

Simplify \sqrt{\dfrac{9}{64}}.

EXPLANATION

Recall that if a\geq 0, then

\sqrt{a^2} = a.

First, notice that 9 and 64 are both perfect squares, where

9=3^2,

and

64 = 8^2.

So, to work out \sqrt{\dfrac{9}{64}}, we express \dfrac{9}{64} as a perfect square:

\begin{align*} \sqrt{\dfrac{9}{64}} &= \\[5pt] \sqrt{\dfrac{3^2}{8^2}} &= \\[5pt] \sqrt{\dfrac{3 \cdot 3}{8 \cdot 8}} &= \\[5pt] \sqrt{\dfrac{3}{8}\cdot \dfrac{3}{8}} &= \\[5pt] \sqrt{\left(\dfrac{3}{8}\right)^2} &= \\[5pt] \dfrac{3}{8}& \end{align*}

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Practice: The Square Root of a Fraction

Simplify $\sqrt{\dfrac{1}{16}}.$

a
 $\dfrac{1}{4}$
b
 $\dfrac{1}{8}$
c
 $4$
d
 $\dfrac{4}{8}$
e
 Undefined
Practice: The Square Root of a Fraction

Simplify $\sqrt{\dfrac{16}{49}}.$

a
 $\dfrac{16}{49}$
b
 $\pm \dfrac{4}{7}$
c
 $\pm \dfrac{4}{49}$
d
 $\pm \dfrac{7}{4}$
e
 $\dfrac{4}{7}$
Practice: The Square Root of a Fraction

Simplify $\sqrt{\dfrac{36}{25}}.$

a
 $\dfrac{6}{5}$
b
 $\dfrac{16}{15}$
c
 $\dfrac{36}{5}$
d
 $\dfrac{6}{25}$
e
 $\dfrac{8}{25}$
Square Roots of Negative Numbers

To work out the square root of a number, we need to express that number as a perfect square. And since the square of any number is always positive or zero, this means that we cannot find the square root of a negative number!

In other words, if we want to find \sqrt{\color{blue}a} , we must have {\color{blue}a} \geq 0.

For now, we'll say that an expression like \sqrt{-1} is not a real number. We'll learn more about real numbers and how to deal with the square root of negative numbers in the future.

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Example: Determining Whether a Square Root Is a Real Number

Which of the following square roots are real numbers?

  1. \sqrt{0}
  2. \sqrt{1.5}
  3. \sqrt{-4}
EXPLANATION

The square root of a negative number is not a real number, while the square root of a non-negative number is a real number.

Let's look at each square root in turn.

  • \sqrt{0} is the square root of 0, which is a non-negative number. So \sqrt{0} is a real number. In fact, since 0 = 0^2, we have \sqrt{0} = \sqrt{0^2} = 0.

  • \sqrt{1.5} is the square root of 1.5, which is a non-negative number. So \sqrt{1.5} is a real number.

  • \sqrt{-4} is the square root of -4, which is a negative number. So \sqrt{-4} is not a real number.

So, the correct answer is "I and II only".

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Practice: Determining Whether a Square Root Is a Real Number

Which of the following square roots are real numbers?

  1. $\sqrt{-196}$
  2. $\sqrt{10}$
  3. $\sqrt{0.25}$
a
 I only
b
 II and III only
c
 II only
d
 I and II only
e
 III only
Practice: Determining Whether a Square Root Is a Real Number

Which of the following square roots are real numbers?

  1. $\sqrt{1}$
  2. $\sqrt{-9}$
  3. $\sqrt{2.4}$
a
 I only
b
 II and III only
c
 III only
d
 I and III only
e
 I and II only
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