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Mathematics LibreTexts

5.7: Simplify and Use Square Roots (Part 2)

Approximate Square Roots with a Calculator

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the \(\sqrt{\text{ }}\) or \(\sqrt{x}\) key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is ≈ and it is read approximately.

Suppose your calculator has a 10-digit display. Using it to find the square root of 5 will give 2.236067977. This is the approximate square root of 5. When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.

$$\sqrt{5} \approx 2.236067978$$

You will seldom use this many digits for applications in algebra. So, if you wanted to round \(\sqrt{5}\) to two decimal places, you would write

$$\sqrt{5} \approx 2.24$$

How do we know these values are approximations and not the exact values? Look at what happens when we square them.

$$\begin{split} 2.236067978^{2} & = 5.000000002 \\ 2.24^{2} & = 5.0176 \end{split}$$

The squares are close, but not exactly equal, to 5.

Example 5.74:

Round \(\sqrt{17}\) to two decimal places using a calculator.

Solution
Use the calculator square root key. 4.123105626
Round to two decimal places. 4.12
  $$\sqrt{17} \approx 4.12$$

Exercise 5.147:

Round \(\sqrt{11}\) to two decimal places.

Exercise 5.148:

Round \(\sqrt{13}\) to two decimal places

Simplify Variable Expressions with Square Roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?

Consider \(\sqrt{9x^{2}}\), where x ≥ 0. Can you think of an expression whose square is 9x2?

$$\begin{split} (?)^{2} & = 9x^{2} \\ (3x)^{2} & = 9x^{2} \\ so\; \sqrt{9x^{2}} & = 3x \end{split}$$

When we use a variable in a square root expression, for our work, we will assume that the variable represents a nonnegative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

Example 5.75:

Simplify: x2.

Solution

Think about what we would have to square to get x2. Algebraically, (?)2 = x2

Since (x)2 = x2 x

Exercise 5.149:

Simplify: \(\sqrt{y^{2}}\).

Exercise 5.150:

Simplify: \(\sqrt{m^{2}}\).

Example 5.76:

Simplify: \(\sqrt{16x^{2}}\).

Solution
Since (4x)2 = 16x2 4x

Exercise 5.151:

Simplify: \(\sqrt{64x^{2}}\).

Exercise 5.152:

Simplify: \(\sqrt{169y^{2}}\).

Example 5.77:

Simplify: \(- \sqrt{81y^{2}}\).

Solution
Since (9y)2 = 81y2 -9y

Exercise 5.153:

Simplify: \(- \sqrt{121y^{2}}\).

Exercise 5.154:

Simplify: \(- \sqrt{100p^{2}}\).

Example 5.78:

Simplify: \(\sqrt{36x^{2} y^{2}}\).

Solution
Since (6xy)2 = 36x2 y2 6xy

Exercise 5.155:

Simplify: \(\sqrt{100a^{2} b^{2}}\).

Exercise 5.156:

Simplify: \(\sqrt{225m^{2} n^{2}}\).

Use Square Roots in Applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

HOW TO: USE A STRATEGY FOR APPLICATIONS WITH SQUARE ROOTS

Step 1. Identify what you are asked to find.

Step 2. Write a phrase that gives the information to find it.

Step 3. Translate the phrase to an expression.

Step 4. Simplify the expression.

Step 5. Write a complete sentence that answers the question.

Square Roots and Area

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.

If the area of the square is A square units, the length of a side is A units. See Table 5.104.

Table 5.104

Area (square units) Length of side (units)
9 $$\sqrt{9} = 3$$
144 $$\sqrt{144} = 12$$
A $$\sqrt{A}$$

Example 5.79:

Mike and Lychelle want to make a square patio. They have enough concrete for an area of 200 square feet. To the nearest tenth of a foot, how long can a side of their square patio be?

Solution

We know the area of the square is 200 square feet and want to find the length of the side. If the area of the square is A square units, the length of a side is \(\sqrt{A}\) units.

What are you asked to find? The length of each side of a square patio
Write a phrase. The length of a side
Translate to an expression. $$\sqrt{A}$$
Evaluate \(\sqrt{A}\) when A = 200. $$\sqrt{200}$$
Use your calculator. 14.142135...
Round to one decimal place. 14.1 feet
Write a sentence. Each side of the patio should be 14.1 feet.

Exercise 5.157:

Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 370 square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?

Exercise 5.158:

Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of 2704 square centimeters. How long can a side of his mosaic be?

Square Roots and Gravity

Another application of square roots involves gravity. On Earth, if an object is dropped from a height of h feet, the time in seconds it will take to reach the ground is found by evaluating the expression \(\frac{\sqrt{h}}{4}\). For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by evaluating \(\frac{\sqrt{64}}{4}\).

Take the square root of 64. $$\frac{8}{4}$$
Simplify the fraction. 2

It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.

Example 5.80:

Christy dropped her sunglasses from a bridge 400 feet above a river. How many seconds does it take for the sunglasses to reach the river?

Solution
What are you asked to find? The number of seconds it takes for the sunglasses to reach the river
Write a phrase. The time it will take to reach the river
Translate to an expression. $$\frac{\sqrt{h}}{4}$$
Evaluate \(\frac{\sqrt{h}}{4}\) when h = 400. $$\frac{\sqrt{400}}{4}$$
Find the square root of 400. 14.142135...
Simplify. 14.1 feet
Write a sentence. Each side of the patio should be 14.1 feet. 

Exercise 5.159:

A helicopter drops a rescue package from a height of 1296 feet. How many seconds does it take for the package to reach the ground?

Exercise 5.160:

A window washer drops a squeegee from a platform 196 feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?

Square Roots and Accident Investigations

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is d feet, then the speed of the car can be found by evaluating \(\sqrt{24d}\).

Example 5.81:

After a car accident, the skid marks for one car measured 190 feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?

Solution
What are you asked to find? The speed of the car before the brakes were applied
Write a phrase. The speed of the car
Translate to an expression. $$\sqrt{24d}$$
Evaluate \(\sqrt{24d}\) when d = 190. $$\sqrt{24 \cdot 190}$$
Multiply. $$\sqrt{4,560}$$
Use your calculator. 67.527772...
Round to tenths. 67.5
Write a sentence. The speed of the car was approximately 67.5 miles per hour.

Exercise 5.161:

An accident investigator measured the skid marks of a car and found their length was 76 feet. To the nearest tenth, what was the speed of the car before the brakes were applied?

Exercise 5.162:

The skid marks of a vehicle involved in an accident were 122 feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

  1. \(\sqrt{36}\) 
  2. \(\sqrt{4}\)
  3. \(\sqrt{64}\)
  4. \(\sqrt{144}\)
  5. \(- \sqrt{4}\)
  6. \(- \sqrt{100}\)
  7. \(- \sqrt{1}\)
  8. \(- \sqrt{121}\)
  9. \(\sqrt{-121}\)
  10. \(\sqrt{-36}\)
  11. \(\sqrt{-9}\)
  12. \(\sqrt{-49}\)
  13. \(\sqrt{9+16}\)
  14. \(\sqrt{25+144}\)
  15. \(\sqrt{9} + \sqrt{16}\)
  16. \(\sqrt{25} + \sqrt{144}\)

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

  1. \(\sqrt{70}\)
  2. \(\sqrt{5}\)
  3. \(\sqrt{200}\)
  4. \(\sqrt{172}\)

Approximate Square Roots with a Calculator

In the following exercises, use a calculator to approximate each square root and round to two decimal places.

  1. \(\sqrt{19}\)
  2. \(\sqrt{21}\)
  3. \(\sqrt{53}\)
  4. \(\sqrt{47}\)

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

  1. \(\sqrt{y^{2}}\)
  2. \(\sqrt{b^{2}}\)
  3. \(\sqrt{49x^{2}}\)
  4. \(\sqrt{100y^{2}}\)
  5. \(- \sqrt{64a^{2}}\)
  6. \(- \sqrt{25x^{2}}\)
  7. \(\sqrt{144x^{2} y^{2}}\)
  8. \(\sqrt{196a^{2} b^{2}}\)

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

  1. Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. How long can a side of his garden be?
  2. Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. How long can a side of his patio be?
  3. Gravity An airplane dropped a flare from a height of 1,024 feet above a lake. How many seconds did it take for the flare to reach the water?
  4. Gravity A hang glider dropped his cell phone from a height of 350 feet. How many seconds did it take for the cell phone to reach the ground?
  5. Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4,000 feet above the Colorado River. How many seconds did it take for the hammer to reach the river?
  6. Accident investigation The skid marks from a car involved in an accident measured 54 feet. What was the speed of the car before the brakes were applied?
  7. Accident investigation The skid marks from a car involved in an accident measured 216 feet. What was the speed of the car before the brakes were applied?
  8. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. What was the speed of the vehicle before the brakes were applied?
  9. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 117 feet. What was the speed of the vehicle before the brakes were applied?

Everyday Math

  1. Decorating Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be?
  2. Decorating Morris wants to have a square mosaic inlaid in his new patio. His budget allows for 2,025 tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?

Writing Exercises

  1. Why is there no real number equal to \(\sqrt{−64}\)?
  2. What is the difference between 92 and \(\sqrt{9}\)?

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

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