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| 16 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
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.
fourth rootof 16. Notice that 416 simplifies to 2 because 2 multiplied by itself 4 times equals 16.
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. |
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.
4is used. For example, the square root of 16 is denoted as
16.
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 | 1=1 | 2 | 2≈1.414213… |
4 | 4=2 | 3 | 3≈1.732050… |
9 | 9=3 | 5 | 5≈2.236067… |
16 | 16=4 | 10 | 10≈3.162277… |
25 | 25=5 | 20 | 20≈4.472135… |
3a3=a and (3a)3=a
In the following applet, the radicands are perfect squares or perfect cubes. With this in mind, calculate the exact square and cube roots.
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 |
---|---|---|
25 | 252 | Rational Number |
18 | 232⋅2 | Irrational Number |
1000 | 2103 | Irrational Number |
327 | 333 | Rational Number |
325 | 352 | Irrational Number |
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.
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.
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.To calculate a square root, push 2ND and x2 on the calculator.
To calculate a cube root, push MATH and go to the fourth option on the calculator.
Tadeo will calculate the exact values of the given irrational numbers by using a calculator. He can start with the first number, 5. Since it a square root, he needs to push 2ND and x2 to have the square root symbol. Then he can enter the value of the radicand, 5, and push ENTER.
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.
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.
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.
Integer | Multiple of 3 | Irrational Number | Rational Number | |
---|---|---|---|---|
3 | ✓ | ✓ | × | ✓ |
6 | ✓ | ✓ | × | ✓ |
11 | ✓ | × | × | ✓ |
All the side lengths are integers. |
All the side lengths are rational numbers. |
All the areas are perfect squares. |
Determine the indices and radicands of the given radical expressions. Write the index first and then the radicand.
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.
Calculate each square root.
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.
Calculate the given cube roots.
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
Calculate the given cube roots.
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
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.
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 |