To square a number means to multiply that number by itself.
For example, to "square
3
3
3.
3
3^2,
3^2 = 3 \cdot 3 = {\color{blue}9}.
3
9
3
9
The number
\color{blue}2
3^{\color{blue}2}
power or exponent .
The act of squaring a number can be represented geometrically. For example, let's start by representing the number
3
Then we can picture
3^2
3\mathbin{:}
There are
9
3^2 = 9.
The first few basic squares are shown below:
1^2=1 \cdot 1={\color{blue}1}
2^2=2 \cdot 2={\color{blue}4}
3^2=3 \cdot 3={\color{blue}9}
4^2=4 \cdot 4={\color{blue}16}
The squares of our usual counting numbers are called square numbers or perfect squares . We can write them as follows:
1^2, 2^2, 3^2, 4^2, 5^2, 6^2, \ldots \qquad \text{or} \qquad {\color{blue}1}, {\color{blue}4}, {\color{blue}9}, {\color{blue}16}, {\color{blue}25}, {\color{blue}36}, \ldots
Calculate the value of
7^2.
To work out
7^2,
7
7^2 = 7 \cdot 7 = 49
a
$169$
b
$12$
c
$140$
d
$170$
e
$144$
To work out $12^2,$ we multiply $12$ by itself:
\[
12^2 =12\cdot 12 = 144
\]
a
$115$
b
$30$
c
$150$
d
$215$
e
$225$
To work out $15^2, $ we multiply $15$ by itself:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \\
& & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \\
\end{array}
%&\qquad\qquad&
%%%% Explanations %%%
%\begin{array}{l}
%69 + 460 = {\color{red} 529}
%\end{array}
%\\[5pt]
%&
\end{align*}
Therefore, $15^2 = 15 \times 15 = 225.$
Find the square of
\dfrac{1}{2}.
To work out
\left(\dfrac 1 2\right)^2,
\dfrac 1 2
\left(\dfrac 1 2\right)^2 = \dfrac 1 2 \cdot \dfrac 1 2 = \dfrac 1 4
$\left( \dfrac{1}{3}\right)^2=$
a
$\dfrac{1}{5}$
b
$\dfrac{1}{9}$
c
$\dfrac{1}{6}$
d
$\dfrac{1}{3}$
e
$\dfrac{2}{3}$
To work out
$\left(\dfrac{1}{3}\right)^2,$
we multiply
$\dfrac{1}{3}$
by itself:
\[
\left(\dfrac{1}{3}\right)^2 = \dfrac{1}{3}\cdot\dfrac{1}{3} = \dfrac{1}{9}
\]
$\left( \dfrac{2}{5}\right)^2=$
a
$\dfrac{8}{25}$
b
$\dfrac{4}{125}$
c
$\dfrac{2}{5}$
d
$\dfrac{8}{125}$
e
$\dfrac{4}{25}$
To work out
$\left(\dfrac{2}{5}\right)^2,$
we multiply
$\dfrac{2}{5}$
by itself:
\[
\left(\dfrac{2}{5}\right)^2 = \dfrac{2}{5}\cdot\dfrac{2}{5} = \dfrac{4}{25}
\]
The squares of negative numbers are always positive because a negative times a negative is positive:
\begin{align*}
\mathbf{\color{red}(-)} \:\mathbf{\cdot}\: \mathbf{\color{red}(-)} \:=\: \mathbf{\color{blue}(+)}
\end{align*}
Some squares of negative numbers are as follows:
\begin{align*}
(-1)^2\:=\:{\color{red}(-1)} \cdot {\color{red}(-1)} &\:=\: {\color{blue}1} \\[5pt]
(-2)^2\:=\:{\color{red}(-2)} \cdot {\color{red}(-2)} &\:=\: {\color{blue}4} \\[5pt]
(-3)^2\:=\:{\color{red}(-3)} \cdot {\color{red}(-3)} &\:=\: {\color{blue}9} \\[5pt]
(-4)^2\:=\:{\color{red}(-4)} \cdot {\color{red}(-4)} &\:=\: {\color{blue}16} \\[5pt]
(-5)^2\:=\:{\color{red}(-5)} \cdot {\color{red}(-5)} &\:=\: {\color{blue}25}
\end{align*}
Watch Out! When squaring negative numbers, the parentheses are important!
Consider the following squared quantity:
-3^2
not the same as
(-3)^2
(-1) \cdot 3^2.
According to the order of operations, we should square
3
first and then multiply by
-1.
\begin{align*}
-3^2
&= (-1)\cdot 3^2 \\[5pt]
&= (-1)\cdot 9 \\[5pt]
&=-9
\end{align*}
What is the value of
(-4)^2?
To work out
(-4)^2,
-4
(-4)^2 = (-4) \cdot (-4) = 16
a
$\pm36$
b
$36$
c
$30$
d
$40$
e
$-36$
To work out $(-6)^2,$ we multiply $-6$ by itself:
\[
\begin{align*}
(-6)^2 =(-6)\cdot (-6) = 36
\end{align*}
\]
a
$115$
b
$225$
c
$-30$
d
$-225$
e
$30$
Since the product of two negative numbers gives a positive number, we have
\[
(-15)^2 = (-15)\cdot(-15) = 15\cdot 15
\]
Next, we calculate $15\cdot 15{:}$
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}\phantom{0}$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \\
& & & \!\!\!\! 1 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\hline
& & \!\!\!\!\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \\
\end{array}
%&\qquad\qquad&
%%%% Explanations %%%
%\begin{array}{l}
%69 + 460 = {\color{red} 529}
%\end{array}
%\\[5pt]
%&
\end{align*}
Therefore, we conclude that
\[
(-15)^2 =15^2 = 225.
\]
What is the square of
-2.5?
We need to calculate the product of
(-2.5)
(-2.5)^2 = (-2.5)\cdot (-2.5) = (2.5)\cdot (2.5)
First, we ignore the decimal points and multiply as if both numbers were whole numbers:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{[math]\color{blue}1[/math]} \\[2pt] \fbox{[math]\color{blue}2[/math]} } \!\!\!\! & \\
& & & \!\!\!\! 2 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}\!\!\!\! & \!\!\!\!\color{red}6\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \\
\end{array}
%&\qquad\qquad&
%%%% Explanations %%%
%\begin{array}{l}
%21 + 420 = {\color{red} 441}
%\end{array}
%\\[5pt]
%&
\end{align*}
Therefore,
25 \times 25 = 625.
We now count the total number of decimal places in the two factors.
There is
\color{blue}1
2.5
\color{blue}1
2.5
{\color{blue}{1}} + {\color{blue}{1}} = 2
We take our value of
625
2
6\,\overset{\color{red}\downarrow}{\color{red}\bbox[2px, lightgray]{.}}\!\!\!\underbrace{25}_{\large\text{[math]2[/math] digits}}
Therefore,
(-2.5)^2 = 6.25\,.
Expressed as a decimal in its simplest form, $(0.2)^2=$
We need to calculate the product of $(0.2)$ with itself:
\[
(0.2)^2 = (0.2)\cdot (0.2)
\]
First, we ignore the decimal points and multiply as if both numbers were whole numbers:
\[
2 \times 2 = 4.
\]
We now count the total number of decimal places in the two factors.
There is $\color{blue}1$ decimal place in the first factor ($0.2$) and $\color{blue}1$ decimal place in the second factor ($0.2$), so their product will have ${\color{blue}{1}} + {\color{blue}{1}} = 2$ decimal places.
We take our value of $4$ and insert a decimal point to make a number with $2$ decimal places:
\[
0\,\overset{\color{red}\downarrow}{\color{red}\bbox[2px, lightgray]{.}}\!\!\!\underbrace{04}_{\large\text{$2$ digits}}
\]
Therefore, $(0.2)^2 = 0.04\,.$
a
$14.5$
b
$12.25$
c
$15$
d
$3.5$
e
$10$
We need to calculate the product of $3.5$ with itself:
\[
3.5^2 = (3.5)\cdot (3.5)
\]
First, we ignore the decimal points and multiply as if both numbers were whole numbers:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}1$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \\
& & & \!\!\!\! 3 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!3\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\hline
& & \!\!\!\!1\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!1\!\!\!\! & \!\!\!\!0\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}1\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \\
\end{array}
%&\qquad\qquad&
%%%% Explanations %%%
%\begin{array}{l}
%175 + 1050 = {\color{red}1 , 225}
%\end{array}
%\\[5pt]
%&
\end{align*}
Therefore, $35 \times 35 = 1\,225.$
We now count the total number of decimal places in the two factors.
There is $\color{blue}1$ decimal place in the first factor ($3.5$) and $\color{blue}1$ decimal place in the second factor ($3.5$), so their product will have ${\color{blue}{1}} + {\color{blue}{1}} = 2$ decimal places.
We take our value of $1\,225$ and insert a decimal point to make a number with $2$ decimal places:
\[
12\,\overset{\color{red}\downarrow}{\color{red}\bbox[2px, lightgray]{.}}\!\!\!\underbrace{25}_{\large\text{$2$ digits}}
\]
Therefore, $3.5^2 = 12.25\,.$
a
$5.5$
b
$30$
c
$30.25$
d
$33$
e
$27.5$
We need to calculate the product of $-5.5$ with itself:
\[
(-5.5)^2 = (-5.5)\cdot (-5.5) = (5.5)\cdot (5.5)
\]
First, we ignore the decimal points and multiply as if both numbers were whole numbers:
\begin{align*}
\require{cancel}
%%%%%%%%%%
%%% Step A %%%
%%%%%%%%%%
&
\begin{array}{ccccc}
& & & \!\!\!\! \color{lightgray} \substack{ \fbox{$\color{blue}2$} \\[2pt] \fbox{$\color{blue}2$} } \!\!\!\! & \\
& & & \!\!\!\! 5 \!\!\!\! & \!\!\!\! 5 \!\!\!\! \\
\!\!\!\!\times\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!\phantom{0}\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\hline
& & \!\!\!\!2\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! \\
\!\!\!\!+\!\!\!\! & \!\!\!\!2\!\!\!\! & \!\!\!\!7\!\!\!\! & \!\!\!\!5\!\!\!\! & \!\!\!\!0\!\!\!\! \\
\hline
& \!\!\!\!\color{red}3\!\!\!\! & \!\!\!\!\color{red}0\!\!\!\! & \!\!\!\!\color{red}2\!\!\!\! & \!\!\!\!\color{red}5\!\!\!\! \\
\end{array}
%&\qquad\qquad&
%%%% Explanations %%%
%\begin{array}{l}
%129 + 1720 = {\color{red}1 , 849}
%\end{array}
%\\[5pt]
%&
\end{align*}
Therefore, $55 \times 55 = 3\,025.$
We now count the total number of decimal places in the two factors.
There is $\color{blue}1$ decimal place in the first factor ($5.5$) and $\color{blue}1$ decimal place in the second factor ($5.5$), so their product will have ${\color{blue}{1}} + {\color{blue}{1}} = 2$ decimal places.
We take our value of $3\,025$ and insert a decimal point to make a number with $2$ decimal places:
\[
30\,\overset{\color{red}\downarrow}{\color{red}\bbox[2px, lightgray]{.}}\!\!\!\underbrace{25}_{\large\text{$2$ digits}}
\]
Therefore, $5.5^2 = 30.25.$
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