# 9.5: Estimation of Square Roots

**At Grade**Created by: CK-12

Have you ever done community service?

Miguel is enjoying his time with the Wildcats. The players have all been very friendly to him and Miguel loves helping out. He gets to hand out towels and water and sometimes collects and organizes equipment.

The Wildcats do a lot of community service. One thing that they do is help out Little League teams. There is one new ball field being created and the Wildcats are going to help design the infield so that it has the correct dimensions. Miguel has been invited to go along.

When they arrive at the field, Miguel and one of the players Harris take a measure to measure the distance from home plate to first base. It measures 58 feet.

“That isn’t correct,” Harris tells Miguel. “The area of the infield for little league is 3600 square feet. This measurement is inaccurate. We’ll have to help them fix it.”

Miguel is puzzled. What is inaccurate about the 58 feet? Does it need to be a longer distance or a shorter one?

**To figure this problem out, Miguel will need to use his knowledge of squares and square roots. Pay attention and you can help Miguel figure this problem out at the end of the lesson.**

### Guidance

In the last Concept, you began learning about ** square roots.** Remember that

**the square root of a number is the number that you multiply by itself to get a product,**in essence it is a number that you square.

**Squaring and square roots are inverse operations.**You can square a number to get a product, then take the square root of the product to get back to the original number.

\begin{align*}6^2 & = 36\\ \sqrt{36} & = 6\end{align*}

**Here you can see the inverse operation between squaring a number and finding the square root of a product.**

**A number like 36 is a** *perfect square***meaning that the square root is a whole number.** Here are some other perfect squares.

16, 25, 36, 49, 64, 81, 100, 121, 144, 169

**Notice that if you find the square root of any of these numbers, you will end up with a whole number as the answer.**

**What if a square root is not a whole number?**

This can also happen. When it does, we can approximate the square root of the number. There are a couple of different ways to do this. Let’s take a look.

**First, we can approximate the square root using perfect squares.**

To do this, we look for the perfect squares that generate a number close to the square root that we are looking for.

\begin{align*}\sqrt{30}\end{align*}

If we are looking for the square root of 30, we first need to find two perfect squares near 30. One should be less than 30 and one should be more than 30.

\begin{align*}5 \times 5 &= 25\\ 6 \times 6 &= 36\end{align*}

25 is the perfect square less than 30.

36 is the perfect square greater than 30.

**Because 30 is between 25 and 30, we can say that the approximate square root of 30 is between 5 and 6. It is probably close to 5.5.**

**This is how we can approximate a square root using perfect squares.**

**The second way of approximating a square root is to use a calculator.** Calculators all have a radical sign on them.

\begin{align*}\sqrt{\Box}\end{align*}

To find the square root of a number, we enter the radical sign, then the value and press enter.

This will give us a decimal approximation of the square root. Many times you will need to round these answers. Let’s look at an example.

We can use 30 again.

\begin{align*}\sqrt{30}\end{align*}

Press the radical button.

Now enter the value, 30.

Next, press enter.

**Here is our answer on the calculator. 5.477225575.**

To find a final answer, we can round to the nearest tenth. To do this, we round the 4 it is in the tenths place. The number after the 4 is 7, so we round up to five.

**Our answer is 5.5.**

**WOW! Notice that this value is the same as the answer that we found when estimating using perfect squares.**

**The third way of approximating a square root is to use something called “tabular interpolation.”** *Tabular interpolation***is using a table.** To use this table, we find the approximate value of the square root according to research that has been completed by a mathematician.

\begin{align*}\sqrt{17}\end{align*}

To find the square root of 17, let’s use a table. Keep in mind that these values have been rounded to the nearest thousandth.

Looking at the table you can see that the square root of 17 is 4.123.

**This is our answer.**

**You can find tables that include numbers all the way up to 100. By finding these tables, you can use them to locate the square root of numbers 1 to 100.**

Choose a method to find an approximate square root of each number. Round your answer to the nearest tenth.

#### Example A

\begin{align*}\sqrt{11}\end{align*}

**Solution: \begin{align*}3.3\end{align*}**

#### Example B

\begin{align*}\sqrt{33}\end{align*}

**Solution:\begin{align*}5.7\end{align*}**

#### Example C

\begin{align*}\sqrt{41}\end{align*}

**Solution:\begin{align*}6.4\end{align*}**

Here is the original problem once again.

Miguel is enjoying his time with the Wildcats. The players have all been very friendly to him and Miguel loves helping out. He gets to hand out towels and water and sometimes collects and organizes equipment.

The Wildcats do a lot of community service. One thing that they do is help out Little League teams. There is one new ball field being created and the Wildcats are going to help design the infield so that it has the correct dimensions. Miguel has been invited to go along.

When they arrive at the field, Miguel and one of the players Harris take a measure to measure the distance from home plate to first base. It measures 58 feet.

“That isn’t correct,” Harris tells Miguel. “The area of the infield for little league is 3600 square feet. This measurement is inaccurate. We’ll have to help them fix it.”

Miguel is puzzled. What is inaccurate about the 58 feet? Does it need to be a longer distance or a shorter one?

**Harris told Miguel that the area of the infield needs to be 3600 square feet for Little League. To figure out what is inaccurate about the current measure of 58 feet, Miguel will need to find the square root of 3600 square feet.**

**Finding the square root will give you the measurement of the distance between the bases. This is the distance that is squared to find the overall area of the infield.**

\begin{align*}\sqrt{3600}\end{align*}

**To find this square root, don’t worry about 3600, drop the zeros and find the square root of 36.**

\begin{align*}6 \times 6 = 36\end{align*}

**Now add back one zero.**

\begin{align*}\sqrt{3600} = 60 \ feet\end{align*}

**The distance between each of the bases should be 60 feet. With a current distance of 58 feet, the distance is short two feet between each base.**

**Miguel is ready to help Harris correct the problem thanks to squares and square roots!**

### Vocabulary

Here are the vocabulary words in this Concept.

- Square Root
- A number multiplied to find a product. The number squared is the square root of the product.

- Perfect Square
- a number whose square root is a whole number.

### Guided Practice

Here is one for you to try on your own.

Approximate this square root to the nearest hundredth.

\begin{align*}\sqrt{65}\end{align*}

**Answer**

To figure this out, we can use the table or the calculator.

Using our calculator, we evaluate this square root as:

\begin{align*}8.0622577\end{align*}

Now we round to the nearest hundredth.

\begin{align*}8.06\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

This is a James Sousa video on approximating square roots. It has a technology integration in it.

### Practice

Directions: Find each an approximate answer for each square root. You may round your decimal answer to the nearest tenth.

1. \begin{align*}{\overline{ ) {8 }}}\end{align*}

2. \begin{align*}{\overline{ ) {11 }}}\end{align*}

3. \begin{align*}{\overline{ ) {24 }}}\end{align*}

4. \begin{align*}{\overline{ ) {31 }}}\end{align*}

5. \begin{align*}{\overline{ ) {37 }}} \end{align*}

6. \begin{align*}{\overline{ ) {43 }}}\end{align*}

7. \begin{align*}{\overline{ ) {59 }}}\end{align*}

8. \begin{align*}{\overline{ ) {67 }}}\end{align*}

9. \begin{align*}{\overline{ ) {73 }}}\end{align*}

10. \begin{align*}{\overline{ ) {80 }}}\end{align*}

11. \begin{align*}{\overline{ ) {95 }}}\end{align*}

12. \begin{align*}{\overline{ ) {99 }}}\end{align*}

13. \begin{align*}{\overline{ ) {101 }}}\end{align*}

14. \begin{align*}{\overline{ ) {150 }}}\end{align*}

15. \begin{align*}{\overline{ ) {136 }}}\end{align*}

### Image Attributions

Here you'll learn to approximate square roots to the nearest given decimal place.