Lesson 5

Reasoning About Square Roots

Let’s approximate square roots.

5.1: True or False: Squared

Decide if each statement is true or false.

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

\(\left(\sqrt{9}\right)^2 = 3\)

\(7 = \left(\sqrt{7}\right)^2\)

\(\left(\sqrt{10}\right)^2 = 100\)

\(\left(\sqrt{16}\right)= 2^2\)

5.2: Square Root Values

What two whole numbers does each square root lie between? Be prepared to explain your reasoning.

  1. \(\sqrt{7}\)
  2. \(\sqrt{23}\)
  3. \(\sqrt{50}\)
  4. \(\sqrt{98}\)

Can we do any better than “between 3 and 4” for \(\sqrt{12}\)? Explain a way to figure out if the value is closer to 3.1 or closer to 3.9.

5.3: Solutions on a Number Line

The numbers \(x\), \(y\), and \(z\) are positive, and \(x^2 = 3\), \(y^2 = 16\), and \(z^2 = 30\).

A numbre line that shows the integers from negative 3 to 9
  1. Plot \(x\), \(y\), and \(z\) on the number line. Be prepared to share your reasoning with the class.
  2. Plot \(\text- \sqrt{2}\) on the number line.


In general, we can approximate the values of square roots by observing the whole numbers around it, and remembering the relationship between square roots and squares. Here are some examples:

  • \(\sqrt{65}\) is a little more than 8, because \(\sqrt{65}\) is a little more than \(\sqrt{64}\) and \(\sqrt{64}=8\).
  • \(\sqrt{80}\) is a little less than 9, because \(\sqrt{80}\) is a little less than \(\sqrt{81}\) and \(\sqrt{81}=9\).
  • \(\sqrt{75}\) is between 8 and 9 (it’s 8 point something), because 75 is between 64 and 81.
  • \(\sqrt{75}\) is approximately 8.67, because \(8.67^2=75.1689\).
A number line with the numbers 8 through 9, in increments of zero point 1, are indicated. 

If we want to find a square root between two whole numbers, we can work in the other direction. For example, since \(22^2 = 484\) and \(23^2 = 529\), then we know that \(\sqrt{500}\) (to pick one possibility) is between 22 and 23.

Many calculators have a square root command, which makes it simple to find an approximate value of a square root.

Video Summary

Glossary Entries

  • irrational number

    An irrational number is a number that is not a fraction or the opposite of a fraction.

    Pi (\(\pi\)) and \(\sqrt2\) are examples of irrational numbers.

  • rational number

    A rational number is a fraction or the opposite of a fraction.

    Some examples of rational numbers are: \(\frac74,0,\frac63,0.2,\text-\frac13,\text-5,\sqrt9\)

  • square root

    The square root of a positive number \(n\) is the positive number whose square is \(n\). It is also the the side length of a square whose area is \(n\). We write the square root of \(n\) as \(\sqrt{n}\).

    For example, the square root of 16, written as \(\sqrt{16}\), is 4 because \(4^2\) is 16.  

    \(\sqrt{16}\) is also the side length of a square that has an area of 16.