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.
| Student Learning Objectives: |
|---|
|
| | 16 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
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.
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Challenge
Finding the Side Lengths of Squares
A backyard garden has square vegetable patches of tomatoes, broccoli, and corn.
a
The tomato patch has an area of 9 square yards. Find the side length of the tomato patch.
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b
The broccoli patch has an area of 36 square yards. Find the length of the side of the broccoli patch.
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c
The corn field has an area of 121 square yards. Find the side length of the corn field.
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d
What are the common characteristics between all the squares?
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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. The notation is made up of a radical symbol, radicand a, and index n.
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.
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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. |
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Pop Quiz
Finding Index and Radicand
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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. The radical symbol sqrt()
is used to indicate the square root of a number. For example, the square root of 16 is written as sqrt(16).
sqrt(16) = 4
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 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 NumbersThe square roots of fractions are calculated by taking square roots of the numerator and denominator separately. 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 first.
0.36=36/100=6^2/10^2 [0.5em]
⇓ sqrt(36/100)=6/10=0.6
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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 NumbersThe cube roots of fractions are calculated by taking cube roots of the numerator and denominator separately. 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 first.
0.027=27/1000=3^3/10^3 [0.5em]
⇓ sqrt(27/1000)=3/10=0.3
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Pop Quiz
Evaluating Roots
Calculate the exact square and cube roots.
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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. 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 \ QThe 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 |
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Pop Quiz
Rational or Irrational?
Determine whether the given number is rational or irrational.
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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.
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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.
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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
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.
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 |
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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
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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
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Pop Quiz
Estimating Roots
Without a calculator, estimate the given square root or cube root to its nearest integer.
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Closure
Finding the Side Lengths of Squares
At the beginning of the lesson, we heard about a backyard garden's vegetable patches.
a
The tomato patch has an area of 9 square yards. Find the side length of the tomato patch.
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b
The broccoli patch has an area of 36 square yards. Find the length of the side of the broccoli patch.
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c
The corn field has an area of 121 square yards. Find the side length of the corn 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
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. | ✓ |
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Roots
Exercise 1.1
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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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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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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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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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