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When we studied exponents in Section
[link] , we noted that
${4}^{2}=16$ and
${\left(-4\right)}^{2}=16.$ We can see that 16 is the square of both 4 and
$-4$ . Since 16 comes from squaring
4 or
$-4$ , 4 and
$-4$ are called the
square roots of 16. Thus 16 has two square roots, 4 and
$-4$ . Notice that these two square roots are opposites of each other.
We can say that
Every positive number has two square roots, one positive square root and one negative square root. Furthermore, the two square roots of a positive number are opposites of each other. The square root of 0 is 0.
The two square roots of 49 are 7 and −7 since
$\begin{array}{lllll}{7}^{2}=49\hfill & \hfill & \text{and}\hfill & \hfill & {\left(-7\right)}^{2}=49\hfill \end{array}$
The two square roots of
$\frac{49}{64}$ are
$\frac{7}{8}$ and
$\frac{-7}{8}$ since
$\begin{array}{lllll}{\left(\frac{7}{8}\right)}^{2}=\frac{7}{8}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\frac{7}{8}=\frac{49}{64}\hfill & \hfill & \text{and}\hfill & \hfill & {\left(\frac{-7}{8}\right)}^{2}=\frac{-7}{8}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\frac{-7}{8}=\frac{49}{64}\hfill \end{array}$
Name both square roots of each of the following numbers.
$\frac{1}{4}$
$\frac{1}{2}\text{\hspace{0.17em}}\text{and\hspace{0.17em}}-\frac{1}{2}$
$\frac{9}{16}$
$\frac{3}{4}\text{\hspace{0.17em}}\text{and\hspace{0.17em}}-\frac{3}{4}$
There is a notation for distinguishing the positive square root of a number $x$ from the negative square root of $x$ .
The horizontal bar that appears attached to the radical sign,
$\sqrt{\begin{array}{c}\\ \end{array}}$ , is a grouping symbol that specifies the radicand.
Because
$\sqrt{x}$ and
$-\sqrt{x}$ are the two square root of
$x$ ,
$$\begin{array}{lllll}\left(\sqrt{x}\right)\left(\sqrt{x}\right)=x\hfill & \hfill & \text{and}\hfill & \hfill & \left(-\sqrt{x}\right)\left(-\sqrt{x}\right)=x\hfill \end{array}$$
Write the principal and secondary square roots of each number.
$$\begin{array}{ccc}9.& & \begin{array}{l}\text{Principal\hspace{0.17em}square\hspace{0.17em}root\hspace{0.17em}is\hspace{0.17em}}\sqrt{9}=3.\\ \text{Secondary\hspace{0.17em}square\hspace{0.17em}root\hspace{0.17em}is}\text{\hspace{0.17em}}-\sqrt{9}=-3.\end{array}\end{array}$$
$$\begin{array}{ccc}15.& & \begin{array}{l}\text{Principal\hspace{0.17em}square\hspace{0.17em}root\hspace{0.17em}is\hspace{0.17em}}\sqrt{15}.\\ \text{Secondary\hspace{0.17em}square\hspace{0.17em}root\hspace{0.17em}is}\text{\hspace{0.17em}}-\sqrt{15}.\end{array}\end{array}$$
Use a calculator to obtain a decimal approximation for the two square roots of 34. Round to two decimal places.
$$\begin{array}{l}\text{On}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{Calculater}\\ \begin{array}{lll}\text{Type}\hfill & \hfill & 34\hfill \\ \text{Press}\hfill & \hfill & \begin{array}{||}\hline \sqrt{x}\\ \hline\end{array}\hfill \\ \text{Display reads:}\hfill & \hfill & 5.8309519\hfill \\ \text{Round to 5.83.}\hfill & \hfill & \hfill \end{array}\end{array}$$
Notice that the square root symbol on the calculator is
$\sqrt{\begin{array}{c}\\ \end{array}}$ . This means, of course, that a calculator will produce only the positive square root. We must supply the negative square root ourselves.
$$\begin{array}{lllll}\sqrt{34}\approx 5.83\hfill & \hfill & \text{and}\hfill & \hfill & -\sqrt{34}\approx -5.83\hfill \end{array}$$
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