第9章平方根と根号

Remember that when a number $n$ is multiplied by itself, we write $n^2$ and read it “n squared.” For example, ${15}^2$ reads as “15 squared,” and 225 is called the square of 15, since ${15}^2=225$.

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is $m$ is called a square root of $m$.

Notice ${\left(\mathrm{-15}\right)}^2=225$ also, so $\mathrm{-15}$ is also a square root of 225. Therefore, both 15 and $\mathrm{-15}$ are square roots of 225.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, $\sqrt{m}$, denotes the positive square root. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because $0^2=0$, $\sqrt{0}=0$. Notice that zero has only one square root.

Since 15 is the positive square root of 225, we write $\sqrt{225}=15$. Fill in Figure 9.2 to make a table of square roots you can refer to as you work this chapter.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write $\sqrt{225}=15$. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $\text{−}\sqrt{225}=\mathrm{-15}$.

When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at Table 9.1 below.

The square roots of numbers between 4 and 9 must be between the two consecutive whole numbers 2 and 3, and they are not whole numbers. Based on the pattern in the table above, we could say that $\sqrt{5}$ must be between 2 and 3. Using inequality symbols, we write: