8.1.5: Reasoning About Square Roots

Lesson

Let's approximate square roots.

Exercise \(\PageIndex{1}\): True or False: Squared

Decide if each statement is true or false.

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

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

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

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

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

Exercise \(\PageIndex{2}\): 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. \(\sqrt{98}\)

Are you ready for more?

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.

Exercise \(\PageIndex{3}\): Solutions on a Number Line

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

Figure \(\PageIndex{1}\)

1. Plot \(x, y\), and \(z\) on the number line. Be prepared to share your reasoning with the class.
2. Plot \(-\sqrt{2}\) on the number line.

Summary

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 more 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 \(8.67^{2}=75.1689\).

Figure \(\PageIndex{2}\): A number line with the numbers 8 through 9, in increments of zero point 1, are indicated. The square root of 64 is indicated at 8. The square root of 65 is indicated between 8 and 8 point 1, where the square root of 65 is closer to 8 point 1. The square root of 75 is indicated between 8 point 6 and 8 point 7, the square root of 75 is closer to 8 point 7. The square root of 80 is indicated between 8 point 9 and 9, where the square root of 80 is closer to 8 point 9. The square root of 81 is indicated at 9.

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.

Glossary Entries

Definition: Irrational Number

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

Pi (\(\pi\)) and \(\sqrt{2}\) are examples of irrational numbers.

Definition: Rational Number

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

Some examples of rational numbers are: \(\frac{7}{4},0,\frac{6}{3},0.2,-\frac{1}{3},-5,\sqrt{9}\)

Definition: 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.

Practice

Exercise \(\PageIndex{4}\)

1. Explain how you know that \(\sqrt{37}\) is a little more than 6.
2. Explain how you know that \(\sqrt{95}\) is a little less than 10.
3. Explain how you know that \(\sqrt{30}\) is between 5 and 6.

Exercise \(\PageIndex{5}\)

Plot each number on the number line: \(6,\sqrt{83},\sqrt{40},\sqrt{64},7.5\)

Figure \(\PageIndex{3}\)

Exercise \(\PageIndex{6}\)

The equation \(x^{2}=25\) has two solutions. This is because both \(5\cdot 5=25\), and also \(-5\cdot -5=25\). So, 5 is a solution, and also -5 is a solution.

Select all the equations that have a solution of -4:

1. \(10+x=6\)
2. \(10-x=6\)
3. \(-3x=-12\)
4. \(-3x=12\)
5. \(8=x^{2}\)
6. \(x^{2}=16\)

Exercise \(\PageIndex{7}\)

Find all the solutions to each equation.

1. \(x^{2}=81\)
2. \(x^{2}=100\)
3. \(\sqrt{x}=12\)

Exercise \(\PageIndex{8}\)

Select all the irrational numbers in the list. \(\frac{2}{3},\frac{-123}{45},\sqrt{14},\sqrt{64},\sqrt{\frac{9}{1}},-\sqrt{99},-\sqrt{100}\)

(From Unit 8.1.3)

Exercise \(\PageIndex{9}\)

Each grid square represents 1 square unit. What is the exact side length of the shaded square?

Figure \(\PageIndex{4}\)

(From Unit 8.1.2)

Exercise \(\PageIndex{10}\)

For each pair of numbers, which of the two numbers is larger? Estimate how many times larger.

1. \(0.37\cdot 10^{6}\) and \(700\cdot 10^{4}\)
2. \(4.87\cdot 10^{4}\) and \(15\cdot 10^{5}\)
3. \(500,000\) and \(2.3\cdot 10^{8}\)

(From Unit 7.3.2)

Exercise \(\PageIndex{11}\)

The scatter plot shows the heights (in inches) and three-point percentages for different basketball players last season.

Figure \(\PageIndex{5}\)

1. Circle any data points that appear to be outliers.
2. Compare any outliers to the values predicted by the model.

(From Unit 6.2.2)