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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Lesson Settings & Tools
| | 16 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Roots
Slide of 16
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.
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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.
Tadeo's father wants him to calculate the side length of each field.
a Find the side length of the corn field.
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b Find the length of the side of the tomato field.
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c Find the side length of the clover field.
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d What are the common characteristics between all the squares?
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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 rootof 16. Notice that sqrt(16) simplifies to 2 because 2 multiplied by itself 4 times equals 16. sqrt(16) = sqrt(2^4) = 2
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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. |
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Finding Index and Radicand
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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
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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 NumbersSeparate 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
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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 NumbersSeparate 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
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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.
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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 \ QIt 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 |
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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.
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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.
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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.
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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.
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.
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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) ≈ 6Discussion
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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) ≈ 5Pop Quiz
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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.
Closure
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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.
Tadeo will calculate the side length of each field.
a What is the side length of the corn field?
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b What is the length of the side of the tomato field?
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c What is the side length of the clover field?
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d What are the common characteristics between all the squares?
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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. |
Roots
Exercise 1.1
Determine the indices and radicands of the given radical expressions. Write the index first and then the radicand.
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We are asked to determine the index and radicand of the given radical expression. Let's recall how to interpret a radical expression. sqrt(a) A radical sign is used to represent the {\color{#FD9000}{n^\text{th}}} root of a number a. The number under the radical sign is called the radicand and the number written in the upper left of the radical sign is called the index. With this in mind, let's now consider the given number. sqrt(32) The number written under the radical sign is 32, so 32 is radicand. The number written in the upper left of the radical sign is 5, which means that the index is 5.
Let's consider the given expression like we did in Part A. sqrt(-119) The number written in the upper left of the radical sign is 7, which represents the index. The number written under the radical sign is -119, which represents the radicand.
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Calculate each square root.
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We want to find the square root of 169. sqrt(169) We need to think of a number that, when raised to the second power, results in 169. This number is 13. 169 &= 13^2 &⇓ sqrt(169) &= 13 Therefore, the positive square root of 169 is 13.
This time we will find the square root of a fraction. sqrt(81/144) Notice that 81 is the square of 9 and 144 is the square of 12. With this in mind, we will now use the inverse of the Power of a Quotient Property.
We ended with the square of a fraction in the radicand. According to the definition of the square root, we can say that the square root of 81144 is 912. 81/144 = (9/12)^2 ⇒ sqrt(81/144)=9/12 Notice that we can also reduce the resulting fraction by 3!
The square root of 81144 can be simplified to 34.
This time we will find the square root of a decimal number. Let's start by rewriting the decimal number as a fraction. 2.56=256/100 Notice that both the numerator and denominator are perfect squares. We know that 256 is the square of 16 and 100 is the square of 10. Let's now rewrite these numbers as powers.
We ended with the square of a decimal number. 2.56 &= 1.6^2 &⇕ sqrt(2.56) &= 1.6 We found that the square root of 2.56 is equal to 1.6.
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Calculate the given cube roots.
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We want to find the cube root of 64. We need to think of a number that results in 64 when multiplied by itself three times. ^3 =64 Since 4 raised to the third power is 64, the number we are looking for is 4. 64 &= 4^3 &⇓ sqrt(64) &= 4
Now we want to look for a number that, when multiplied by itself three times, results in -216. Notice that the radicand is negative and the index 3 is odd. This means that the result will be negative too. ^3=-216 We know that -6 raised to the third power is -216. Then, the cube root of -216 is -6. -216 &= (-6)^3 &⇓ sqrt(-216) &= -6
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Calculate the given cube roots.
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We want to find the cube root of a fraction. Let's start by thinking of numbers that give us the values in the numerator and denominator when multiplied by themselves three times. ^3=343 ^3=512 Notice that these numbers are perfect cubes because 343 is the third power of 7 and 512 is the third power of 8. 7^3=343 8^3=512 Let's now use these numbers to rewrite the fraction in the radicand.
Since 78 multiplied by itself three times gives 343512, we can say that the cube root of 343512 is 78. 343/512=(7/8)^3 ⇒ sqrt(343/512)=7/8
This time we will look for a number that, when multiplied by itself three times, results in -0.125. Notice that the radicand is a negative decimal number. ^3=-0.125 Resulting a negative number when the number is multiplied by itself three times means that the number itself is also negative. Let's rewrite the decimal number in the radicand as a fraction. -0.125=- 125/100 Great! We will now consider the numerator and denominator separately like we did in Part A.
We found that -0.5 multiplied by itself three times gives -0.125. This means that the cube root of -0.125 is -0.5. -0.125= (-0.5)^3 ⇒ sqrt(-0.125)=-0.5
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Determine which of the following numbers are irrational. Select all that apply.
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We will examine the given numbers one by one to determine which are irrational numbers. Let's start by recalling some characteristics of irrational numbers.
- Irrational numbers cannot be written as a ratio of two integers.
- A number is irrational if it is not rational.
- The decimal expansion of irrational numbers is not repeating and non-terminating.
- The square root of any whole number that is not a perfect square is irrational.
- The cube root of any integer that is not a perfect cube is irrational.
Let's start with the famous number π and take a look at its decimal expansion. π=3.14159265359 ... ✓ Since it is not repeating and non-terminating, we can say that it is irrational. Next, let's have a look at sqrt(16). Note that 16 is a perfect square. 16 = 4^2 ⇒ sqrt(16) = 4 * Since 4 is a rational number, we have that sqrt(16) is not irrational. Now let's consider sqrt(216). Note that 216 is a perfect cube. 216 = 6^3 ⇒ sqrt(216) = 6 * Once again we ended with a rational number, which means that sqrt(216) is not irrational. Let's now look at the decimal number 0.32. It is a rational number because it is a terminating decimal. We can also express it as a ratio. 0.32 = 32/100 * Then, we will consider sqrt(32). Notice that 32 is not a perfect square, meaning that we cannot remove the square root sign. As a result, sqrt(32) is irrational. sqrt(32) ✓ Moving on to sqrt(12), note that 12 is not a perfect cube. This means that sqrt(12) is also irrational. sqrt(12) ✓ Last is -sqrt(12). Notice that 12 cannot be rewritten as a number squared, so we cannot remove the square root sign. This means that -sqrt(12) is irrational. -sqrt(1/2) ✓ Finally, let's make a table to see which numbers are irrational and which numbers are rational.
| Number | Type of the Number |
|---|---|
| π | Irrational |
| sqrt(32) | Irrational |
| sqrt(12) | Irrational |
| -sqrt(1/2) | Irrational |
| sqrt(216) | Rational |
| sqrt(16) | Rational |
| 0.32 | Rational |
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