9.1 Square root expressions

Overview

• Square Roots
• Principal and Secondary Square Roots
• Meaningful Expressions
• Simplifying Square Roots

Square roots

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

Square root

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.

Sample set a

Practice set a

Name both square roots of each of the following numbers.

Principal and secondary square roots

There is a notation for distinguishing the positive square root of a number $x$ from the negative square root of $x$ .

Principal square root: $\sqrt{x}$

Secondary square root: $-\sqrt{x}$

Radical sign, radicand, and radical

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 $x$ ,

$$\begin{array}{lllll}\left(\sqrt{x}\right)\left(\sqrt{x}\right)=x\hfill & \hfill & \text{and}\hfill & \hfill & \left(-\sqrt{x}\right)\left(-\sqrt{x}\right)=x\hfill \end{array}$$

Sample set b

Write the principal and secondary square roots of each number.