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2. Roots
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Chapter 5
2. 

Roots

This lesson focuses on the foundational concepts of square roots and cube roots, as well as the properties of irrational numbers. It explains how to calculate roots and what radicands and indices are in this context. The lesson is particularly useful for students who are looking to deepen their understanding of these mathematical concepts. For example, knowing how to work with roots can help in solving real-world problems like calculating areas and volumes. Understanding irrational numbers can also be beneficial for various scientific applications. The teaching approach combines theoretical knowledge with practical examples, making it easier to grasp these complex topics.

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Student Learning Objectives:
  • Finding the roots of real numbers
  • Identify irrational numbers
  • Estimate roots
16 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Roots
Slide of 16
Squaring or cubing a number are operations related to powers. This lesson will introduce how to undo these operations by using roots, particularly square roots and cube roots. We will also explore irrational numbers.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Challenge

Finding the Side Lengths of Squares

A backyard garden has square vegetable patches of tomatoes, broccoli, and corn.

Tomato patch 9 sqyd, broccoli patch 36 sqyd, corn 121 sqyd
a

The tomato patch has an area of 9 square yards. Find the side length of the tomato patch.

b

The broccoli patch has an area of 36 square yards. Find the length of the side of the broccoli patch.

c

The corn field has an area of 121 square yards. Find the side length of the corn field.

d

What are the common characteristics between all the squares?

Discussion

n^(th) Root

The n^(th) root of a real number a expresses another real number that, when multiplied by itself n times, will result in a. In addition to the radical symbol, the notation is made up of the radicand a and the index n.

The resulting number is commonly called a radical. For example, the radical expression sqrt(16) is the fourth root of 16. Notice that sqrt(16) simplifies to 2 because 2 multiplied by itself 4 times equals 16. sqrt(16) = sqrt(2^4) = 2


Just as with exponents, the most common roots have special names: square roots and cube roots have an index of 2 and 3, respectively.

For example, the radical expression sqrt(16) is the fourth root of 16. Notice that sqrt(16) simplifies to 2 because 2 multiplied by itself 4 times equals 16.

sqrt(16) = sqrt(2^4) = 2
Discussion

Principal Root

Even n^\text{th} roots are defined only for non-negative real numbers. When n is even and the radicand is positive, two real roots exist; the positive one is known as the principal root. For example, the number 16 has two square roots. ccc & &sqrt(16)=± 4& & &⇕& &4^2=16 &and& (-4)^2=16 The principal root of 16 is 4. In contrast, when considering odd roots, such as cube roots, there is only one real root, which then is the principal root. sqrt(27)=3 ⇔ 3^3=27 Since odd roots are defined for all real numbers, the principal root can be either negative or positive. Suppose that n is an integer greater than 1 and a is a real number.

n is even n is odd
a> 0 Two square roots: one positive and one negative. The principal root is the positive one. One positive root, which is the principal root.
a= 0 The only root is zero. The only root is zero.
a< 0 No real roots. One negative root, which is the principal root.
Pop Quiz

Finding Index and Radicand

Identify the index and radicand of the given root.

Random roots
Discussion

Commonly Used Roots

The most common roots have an index of 2 or 3. These common roots have special names: square roots and cube roots, respectively.

Concept

Square Root

A square root of a number a is a number that, when multiplied by itself, equals a. For example, 4 and - 4 are the square roots of 16. 4* 4 &= 16 [0.5em] -4 * (-4)&=16 All positive numbers have two square roots — one positive and one negative. To avoid ambiguity, when talking about the square root of a number, only the positive root, also known as its principal root, is considered. Furthermore, to denote the square root, the symbol sqrt() is used. For example, the square root of 16 is denoted as sqrt(16). sqrt(16) = 4 In the example above, the principal root of 16 is 4 because 4 multiplied by itself equals 16 and 4 is positive. When a number is a perfect square, its square roots are integers.

Principal Root of Perfect Squares Principal Root of Non-Perfect Squares
Perfect Square Principal Root
(Integer Number) 		Non-Perfect Square Principal Root
(Irrational Number)
1 sqrt(1)=1 2 sqrt(2)≈ 1.414213...
4 sqrt(4)=2 3 sqrt(3)≈ 1.732050...
9 sqrt(9)=3 5 sqrt(5)≈ 2.236067...
16 sqrt(16)=4 10 sqrt(10)≈ 3.162277...
25 sqrt(25)=5 20 sqrt(20)≈ 4.472135...

The square root of a negative number is not a real number. This is because there is no real number that, when multiplied by itself, results in a negative number. a * a ≥ 0, for any real numbera


Extra

 Square Roots of Fractions and Decimal Numbers
Separate from whole numbers, the square roots of fractions can be calculated by taking square roots of the numerator and denominator separately. Consider the following example. 9/16=3^2/4^2 ⇒ sqrt(9/16)=3/4 The square roots of decimal numbers can be calculated by writing them in the fraction form. Then, the square roots of the numerator and denominator are calculated. Consider the following example. 0.36=36/100=6^2/10^2 [0.5em] ⇓ sqrt(36/100)=6/10=0.6
Discussion

Cube Root

The cube root of a number a is a number that, when multiplied by itself three times, equals a. For example, 2 is the cube root of 8. 2 * 2 * 2 = 8 ⇒ sqrt(8) = 2 Cube roots and raising a number to the third power are operations that undo each other.

sqrt(a^3) = a and (sqrt(a))^3=a

Unlike square roots, cube roots can yield negative results. This is because the product of three negative factors is negative. ( - 2)( - 2)( - 2)=- 8 ⇒ sqrt(-8) = -2

Extra

 Cube Root of Fractions and Decimal Numbers
Separate from the integers, the cube roots of fractions can be calculated by taking cube roots of the numerator and denominator separately. Consider the following example. 8/125=2^3/5^3 ⇒ sqrt(8/125)=2/5 The cube roots of decimal numbers can be calculated by writing them in the fraction form, calculating the cube roots of the numerator and denominator separately. Consider the following example. 0.027=27/1000=3^3/10^3 [0.5em] ⇓ sqrt(27/1000)=3/10=0.3
Pop Quiz

Evaluating Roots

Calculate the exact square and cube roots.

Random square and cube roots.
Discussion

Irrational Numbers

The set of irrational numbers is formed by all numbers that cannot be expressed as the ratio between two integers. sqrt(2), sqrt(3), sqrt(5), e, π Irrational numbers are real numbers, but they cannot be expressed as fractions. Also, the decimal expansion of irrational numbers is not repeating and non-terminating. sqrt(2) &= 1.41421356237... π &= 3.14159265359 ... In other words, a number is irrational if it is not rational. Although this number set does not have its own symbol, it is sometimes represented with a combination of other symbols.

R-Q or R \ Q

The square root of a whole number that is not a perfect square is irrational. The cube root of an integer that is not a perfect cube is also irrational. For other roots, check whether the radicand is a perfect power that matches the index of the root.

Radical Number Rewrite Result
sqrt(25) sqrt(5^2) Rational Number
sqrt(18) sqrt(3^2* 2) Irrational Number
sqrt(27) sqrt(3^3) Rational Number
sqrt(25) sqrt(5^2) Irrational Number
Pop Quiz

Rational or Irrational?

Determine whether the given number is rational or irrational.

Random Numbers
Discussion

Is sqrt(2) an Irrational Number?

Consider taking the square root of a number. If the radicand is a perfect square, then it is a rational number. If the radicand is not a perfect square, then it is an irrational number. The radicand of sqrt(2) is 2. 2 is not a perfect square ⇓ sqrt(2) is an irrational number We can use a calculator to find the exact value of sqrt(2). To write the square root sign sqrt(), push 2ND and x^2. Then, press the 2 and ENTER buttons.

Although a finite number of decimals is displayed on the calculator screen, the truth is that this value does not terminate. As such, its exact form is sqrt(2). The values of irrational numbers are often rounded to one or two decimal places to easily locate them on a number line. 1.414213562 ... ≈ 1.4 The value of sqrt()2 is less than 2 and greater than 1. Since 1.4 is close to 1.5, the value is located around 1.5 on the number line.

number line
Example

Walking the Trails

A park has five walking trails. Their lengths, in kilometers, are represented by the following irrational numbers.

Trail Length (km)
Robin Trail sqrt(5)
Meadowlark Trail sqrt(5)
Cardinal Trail sqrt(14)
Hummingbird Trail sqrt(20)
Owl Trail sqrt(34)

Use a calculator to find the approximate length of each trail. Then, order these lengths from least to greatest.

Hint
To calculate a square root, push 2ND and x^2 on the calculator. To calculate a cube root, push MATH and go to the fourth option on the calculator.

Solution
We start with the length of the Robin Trail, sqrt(5) kilometers. To find it, we push 2ND and x^2 to enter the radical symbol. Then, we enter the value of the radicand, 5, and push ENTER.

The Meadowlark Trail has a length of sqrt(5) kilometers. Since this is a cube root, we push the MATH button first, then select the fourth option sqrt()(. We enter the value of the radicand, 5, and push the ENTER button to see the result.

We use the same process to find the values of the remaining radicals.

Next, we round each radical to two decimal places.

Trail Length (km) Rounding
Robin Trail sqrt(5) = 2.236067977 ... 2.24
Meadowlark Trail sqrt(5) = 1.709975946 ... 1.71
Cardinal Trail sqrt(14) = 2.410142264 ... 2.41
Hummingbird Trail sqrt(20) = 4.472135955 ... 4.47
Owl Trail sqrt(34) = 3.239611801 ... 3.24

Finally, let's sort these numbers from least to greatest using a number line.

Five rational numbers expressed in radical form were plotted on a number line to determine their magnitude order.

To finish, we can write the lengths of the walking trails in order from shortest to longest in the table.

Trail Exact Length (km) Estimated Length (km)
Meadowlark Trail sqrt(5) 1.71
Robin Trail sqrt(5) 2.24
Cardinal Trail sqrt(14) 2.41
Owl Trail sqrt(34) 3.24
Hummingbird Trail sqrt(20) 4.47
Discussion

Rounding Square Roots

A square root can be rounded to its nearest whole number without calculating its exact value. To do so, start by finding the closest perfect squares to the radicand and calculate their square roots. The initial square root will be located between the two consecutive whole numbers. For example, consider sqrt(40) and several perfect squares close to 40. 25 , 36 , 49 , 64 Since 40 is greater than 36 and less than 49, sqrt(40) is between the square roots of these two values. c 36 & < & 40 & < & 49 sqrt(36) & < & sqrt(40) & < & sqrt(49) Now calculate the square roots of the two perfect squares. c sqrt(36) & < & sqrt(40) & < & sqrt(49) 6 & < & sqrt(40) & < & 7 It can be concluded that the exact value of sqrt(40) is between 6 and 7. Next, we will round the value to one of these numbers. Which number is closer to sqrt(40), 6 or 7? 40- 36=4 & 49-40=9 ⇓ 4 < 9 Since 40 is closer to 36 than to 49, sqrt(40) is closer to sqrt(36) than it is to sqrt(49). Because of this, sqrt(40) rounds to sqrt(36), or 6.

sqrt(40) ≈ 6
Discussion

Rounding Cube Roots

A cube root can be rounded to the nearest integer without using a calculator. For example, consider sqrt(100). sqrt(100) First, find the perfect cubes closest to 100. The greatest perfect cube less than 100 is 64, and the least perfect cube greater than 100 is 125. 64 & < & 100 & < & 125 The inequality means that the cube root of 100 is between the cube roots of 64 and 125. sqrt(64) < sqrt(100) < sqrt(125) Next, calculate the cube roots of the two perfect cubes. 4 & < & sqrt(100) & < & 5 The exact value of sqrt(100) is somewhere between 4 and 5. Next, determine which number is closer to sqrt(100). 100- 64 = 36 & 125-100=25 ⇓ 36 > 25 Since 100 is closer to 125 than it is to 64, sqrt(100) is closer to sqrt(125) than to sqrt(64). Therefore, we round it to the integer 5.

sqrt(100) ≈ 5
Pop Quiz

Estimating Roots

Without a calculator, estimate the given square root or cube root to its nearest integer.

Random square and cube roots
Closure

Finding the Side Lengths of Squares

At the beginning of the lesson, we heard about a backyard garden's vegetable patches.

Tomato patch 9 sqyd, broccoli patch 36 sqyd, corn 121 sqyd
a

The tomato patch has an area of 9 square yards. Find the side length of the tomato patch.

b

The broccoli patch has an area of 36 square yards. Find the length of the side of the broccoli patch.

c

The corn field has an area of 121 square yards. Find the side length of the corn field.

d

What are the common characteristics between all the squares?

Hint
a

The area of a square is the square of a side length. Squaring a number and taking the square root of the number are inverse operations.

b

The area of a square is the square of a side length. Squaring a number and taking the square root of the number are inverse operations.

c

The area of a square is the square of a side length. Squaring a number and taking the square root of the number are inverse operations.

d

Check each side length and area based on the given characteristics.

Solution
a

A square is a rectangle with four equal side lengths. We can find the area of a square by squaring the length of a side.

c Side Length & Area a & a^2 The tomato patch is a square, so we can use the inverse operation to find its side length. We take the square root of the area. c Area & Side Length a^2 & sqrt(a^2)=a We know that 9 is a perfect square because it can be written as the square of 3. Let's take the square root of 9 to find the side length of the tomato patch. c Area & Side Length 9 & sqrt(3^2)=3 The side length of the tomato patch is 3 yards.

b

To find the side length of the broccoli patch, we use the same method as in Part A. We take the square root of 36 to undo the squaring.

c Area & Side Length [0.5em] a^2 & sqrt(a^2)=a [0.5em] 36 & sqrt(6^2)=6 The side length of the broccoli patch is 6 yards. This also means that 36 is a perfect square.

c

We continue with the last field. This time, we take the square root of 121. Because 11^2 = 121, 121 is also a perfect square.

c Area & Side Length [0.5em] a^2 & sqrt(a^2)=a [0.5em] 121 & sqrt(11^2)=11 The side length of the corn field is 11 yards.

d

Lastly, we need to find what the areas and side lengths have in common. In the previous parts, we found that all the areas are perfect squares. This also tells us that all the side lengths are integers.

Perfect Square? Area Side Length Integer?
✓ 9yd 3yd^2 ✓
✓ 36yd 6yd^2 ✓
✓ 121yd 11yd^2 ✓

Since every integer is a rational number, each side length is also a rational number, not an irrational number. Let's check whether the side lengths are multiples of 3.

Multiple of 3
3 ✓
6 ✓
11 *

We found that three of the given characteristics are common to all the vegetable patches.

All the side lengths are integers. ✓
All the side lengths are rational numbers. ✓
All the areas are perfect squares. ✓



Level 1
Level 2
Level 3
Roots
Exercise 3.1

The following figure is made up of five identical squares.

Five Squares

The area of each square is 36 square centimeters. What is the perimeter of this figure?

We are given that the area of each square is 36 square centimeters. We want to find the perimeter of the figure, which means that we need to find the side length of one of the squares. Remember that we can find the area of a square by finding the square of one side length. cc Side Length & Area a & a^2 Since we know the area of a square, we can find one side length of a square by taking the square root of the area. cc Area & Side Length a^2 & sqrt(a^2)=a Now let's calculate the square root of 36. Notice that 36 is a perfect square.

The side length of one square is 6 centimeters. Since the squares are identical, that means that the side length of all of the squares is 6 centimeters! Let's see all the side lengths on the figure!

As shown, there are twelve equal sides of the figure in total. Now we will add them up, or we can multiply 12 times 6 centimeters to find the perimeter of the figure. 12 * 6 = 72 The perimeter of the figure is 72 centimeters.

E
Hint & Answer
S
Solution

Determine when the following sentence is true.

If a certain number is a perfect square and also a perfect cube, then its square root and cube root are the same number.

We know that a certain number is both a perfect square and a perfect cube. We will determine whether it is possible that its square root and its cube root be the same number and how often this happens. Let's recall the definition of a perfect square first.

Perfect Square |- A perfect square is a number that is the square of an integer.

Now let's remember the definition of a perfect cube.

Perfect Cube |- A perfect cube is a number that is a cube of an integer.

Let's list the perfect squares and the perfect cubes that are between 1 and 100.

Perfect Squares Perfect Cubes
1, 4, 9, 16, 25, 36, 49, 64, 81, 100 1, 8, 27, 64

We can see that the numbers 1 and 64 are both perfect squares and perfect cubes. 1 &= 1 * 1 = 1 * 1 * 1 64 &= 8 * 8 = 4 * 4 * 4 Let's calculate the square roots and cube roots of these numbers. c Square Root & Cube Root [0.3em] sqrt(1) = 1 & sqrt(1) = 1 [0.3em] sqrt(64) = 8 & sqrt(64) = 4 We can see that for 1, the square root and the cube root are the same. However, for 64, the square root and cube root are different. Therefore, the square root and cube root of a number that is a perfect square and perfect cube at the same time are sometimes the same number.

E
Hint & Answer
S
Solution

Roots

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