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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. |