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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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Roots
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Taking the square or taking the cube of a number are operations related to powers. This lesson will introduce how to undo these two operations by using roots, particularly square roots and cube roots. In this context, the lesson 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

Tadeo's father is a farmer and he grows corn, tomatoes, and clover. His cultivated fields are in the shape of a square. The areas of these fields are written in the center of the following squares.

Three Square Fields

Tadeo's father wants him to calculate the side length of each field.

a Find the side length of the corn field.
b Find the length of the side of the tomato field.
c Find the side length of the clover 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
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 such as square root and cube root, respectively. These roots will be introduced here, starting with the square root.

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

In the following applet, the radicands are perfect squares or perfect cubes. With this in mind, 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

It is worth knowing that the square root of any whole number that is not a perfect square is irrational. Similarly, the cube root of any integer that is not a perfect cube is irrational. Therefore, to determine whether the root of an integer number is irrational, check the index of the root and the power of the radicand. Consider the following examples.

Radical Number Rewrite Result
sqrt(25) sqrt(5^2) Rational Number
sqrt(18) sqrt(3^2* 2) Irrational Number
sqrt(1000) sqrt(10^3) Irrational Number
sqrt(27) sqrt(3^3) Rational Number
sqrt(25) sqrt(5^2) Irrational Number
Pop Quiz

Irrational Number or Not?

A real number cannot be both a rational number and an irrational number at the same time. It must belong to one or the other of these number sets. Determine whether the given number is rational or irrational.

Random Numbers
Discussion

Is sqrt(2) an Irrational Number?

Numbers that are not rational are called irrational numbers. 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. Note that the radicand of sqrt(2) is 2. 2 is not a perfect square ⇓ sqrt(2) is an irrational number A calculator can be used 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 and instead has infinite decimals. Therefore, 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 Note that 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

Finding the Amounts of Seeds

Tadeo's father wants to store five different seeds to plant next year. He tells Tadeo the amount of seeds he has in kilograms by using the following irrational numbers.

There are five sacks, each labeled with its weight in kilograms using an irrational number in root form. The sacks contain different seeds.
He wants Tadeo to calculate the exact values of the amount of seeds by using a calculator. Help him find the values and round the results to two decimal places. Then, order these amounts 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

Tadeo will calculate the exact values of the given irrational numbers by using a calculator. He can start with the first number, sqrt(5). Since it a square root, he needs to push 2ND and x^2 to have the square root symbol. Then he can enter the value of the radicand, 5, and push ENTER.

Now, round the value to two decimal places for convenience. sqrt(5) = 2.236067977... ≈ 2.24 Next, he will find the exact value of sqrt(5). Since it is a cube root, he needs to push MATH button first, then go to the fourth option by pushing 4. After that, he can enter the value of the radicand, 5, and push ENTER button to see the result.

Now he can round the exact value to two decimal places. sqrt(5) = 1.709975946... ≈ 1.71 Tadeo can apply the same processes to get the values of the remaining radicals.

Finally, Tadeo will round these three exact values to two decimal places. sqrt(14)=2.410142264... ≈ 2.41 sqrt(20)=4.472135955... ≈ 4.47 sqrt(34)=3.239611801... ≈ 3.24 The seed bags contain 2.24, 1.71, 2.41, 4.47, and 3.24 kilograms, respectively. Now he will sort these numbers from least to greatest. 1.71 < 2.24 < 2.41 < 3.24 < 4.47 ⇓ sqrt(5) < sqrt(5) < sqrt(14) < sqrt(34) < sqrt(20) Finally, he wants to locate the rounded values on a number line to see the order.

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

Tadeo's father is so glad that Tadeo completed the task, he will allow Tadeo to move the bags of seeds into storage. Tadeo laughs weakly and starts the chore.

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. Their square roots are then calculated. 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 Note that 40 is greater than 36 and less than 49. Therefore, 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, the value will be rounded to one of these numbers. Which number is closer to sqrt(40)? 40- 36=4 & 49-40=9 ⇓ 4 < 9 Since 40 is closer to 36 than 49, sqrt(40) is closer to sqrt(36) than sqrt(49). Therefore, sqrt(40) rounds to sqrt(36), which is equal to 6.

sqrt(40) ≈ 6
Discussion

Rounding Cube Roots

A cube root can be rounded to its nearest integer without using a calculator. Consider, for example, sqrt(100). First, find the closest perfect cubes to 100. Note that 125 is the least perfect cube greater than 100 and 64 is the greatest perfect cube less than 100. 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 As shown, the exact value of sqrt(100) is between 4 and 5. Now, it will be decided which number is closer to sqrt(100). 100- 64 = 36 & 125-100=25 ⇓ 36 > 25 Since 100 is closer to 125 than 64, sqrt(100) is closer to sqrt(125) than sqrt(64). Therefore, it can be rounded 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. The given radicands are not perfect squares or perfect cubes, so start by thinking about the nearest perfect squares or perfect cubes to the radicands.

Random square and cube roots
Closure

Finding the Side Lengths of Squares

At the beginning of the lesson, it was said that Tadeo's father is a farmer that grows corn, tomatoes, and clover. The cultivated fields are squares.

Three Square Fields

Tadeo will calculate the side length of each field.

a What is the side length of the corn field?
b What is the length of the side of the tomato field?
c What is the side length of the clover 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 Recall that a square as a geometric figure is a rectangle with four equal side lengths. The area of a square can be found by finding the square of a side length.

c Side Length & Area a & a^2 The numbers written in the centers of the cultivated fields are the squares of their corresponding side lengths. An inverse operation can be applied to find the side lengths. To undo the operation of squaring, calculate the square root of each area. c Area & Side Length a^2 & sqrt(a^2)=a Note that 9 is a perfect square because it can be written as the square of 3. With this in mind, now take the square root of 9 to find the side length of the given field. c Area & Side Length 9 & sqrt(3^2)=3 The side length of the corn field is 3 yards.

b Apply the same process from Part A to calculate the side length of the tomato field. Take square root of 36 to undo the operation of squaring. Since 36 is the square of 6, it is a perfect square.

c Area & Side Length a^2 & sqrt(a^2)=a 36 & sqrt(6^2)=6 So, the side length of the tomato field is 6 yards.

c The side length of the last field, the clover field, will be calculated now. Note that 121 is a perfect square because it is the square of 11. To find the side length of the field, take the square root of 121.

c Area & Side Length a^2 & sqrt(a^2)=a 121 & sqrt(11^2)=11 The side length of the clover field is 11 yards.

d This time, it is asked to find the common characteristics between all the squares. The side lengths will be first examined in a table to check whether they are integers, multiples of 3, irrational numbers, or rational numbers.
Integer Multiple of 3 Irrational Number Rational Number
3 ✓ ✓ * ✓
6 ✓ ✓ * ✓
11 ✓ * * ✓

According to the table, all the side lengths are integers and rational numbers, but not all are multiples of 3 or irrational numbers. As a final option, consider whether all the areas are perfect squares. c Area & Perfect Square? 9 & 9 = 3^2 ✓ 36 & 36 = 6^2 ✓ 121 & 121 = 11^2 ✓ All the areas are perfect squares because all of them are the square of an integer. It can be concluded that three of the given options are common characteristics between all the squares.

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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