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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.

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

## Root

The root of a real number expresses another real number that, when multiplied by itself times, will result in In addition to the radical symbol, the notation is made up of the radicand and the index
The resulting number is commonly called a radical. For example, the radical expression is the fourth root of Notice that simplifies to because multiplied by itself times equals
Discussion

## Principal Root

Even roots are defined only for non-negative real numbers. When is even and the radicand is positive, two real roots exist; the positive one is known as the principal root. For example, the number has two square roots.
The principal root of is In contrast, when considering odd roots, such as cube roots, there is only one real root, which then is the principal root. Since odd roots are defined for all real numbers, the principal root can be either negative or positive. Suppose that is an integer greater than and is a real number.
is even is odd
Two square roots: one positive and one negative. The principal root is the positive one. One positive root, which is the principal root.
The only root is zero. The only root is zero.
No real roots. One negative root, which is the principal root.
Pop Quiz

Identify the index and radicand of the given root.

Discussion

## Commonly Used Roots

The most common roots have an index of or 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 is a number that, when multiplied by itself, equals For example, and are the square roots of
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 is used. For example, the square root of is denoted as
In the example above, the principal root of is because multiplied by itself equals and 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)
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.

### 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.
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.
Discussion

## Cube Root

The cube root of a number is a number that, when multiplied by itself three times, equals For example, is the cube root of
Cube roots and raising a number to the third power are operations that undo each other.

Unlike square roots, cube roots can yield negative results. This is because the product of three negative factors is negative.

### 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.
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.
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.

Discussion

## Irrational Numbers

The set of irrational numbers is formed by all numbers that cannot be expressed as the ratio between two integers.
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.
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.

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.

Rational Number
Irrational Number
Irrational Number
Rational Number
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.

Discussion

## Is 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 is
A calculator can be used to find the exact value of To write the square root sign push and Then, press the and 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 The values of irrational numbers are often rounded to one or two decimal places to easily locate them on a number line.
Note that the value of is less than and greater than Since is close to the value is located around on the 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.

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 and on the calculator.

To calculate a cube root, push 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, Since it a square root, he needs to push and to have the square root symbol. Then he can enter the value of the radicand, and push

Now, round the value to two decimal places for convenience.
Next, he will find the exact value of Since it is a cube root, he needs to push button first, then go to the fourth option by pushing After that, he can enter the value of the radicand, and push button to see the result.
Now he can round the exact value to two decimal places.
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.
The seed bags contain and kilograms, respectively. Now he will sort these numbers from least to greatest.
Finally, he wants to locate the rounded values on a number line to see the order.