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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.

Powers
Rounding a Number
Rational Number
Integers
Decimal Number
Perfect Square
Perfect Cube

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

$n^{th}$ 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

416

is the fourth root of

16 .

Notice that

416

simplifies to

2

because

2

multiplied by itself

4

times equals

16 .

416 = 4 2^{4} = 2

Discussion

Principal Root

Even

n^{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.

4^{2} = 16 16 = \pm 4 ⇕ and (- 4)^{2} = 16

The principal root of

16

4 .

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

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.

Pop Quiz

Finding Index and Radicand

Identify the index and radicand of the given root.

Discussion

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

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 \cdot 4 - 4 \cdot (- 4) = 16 = 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

4

is used. For example, the square root of

16

is denoted as

16 .

16 = 4

In the example above, the principal root of

16

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)
$1$	$1 = 1$	$2$	$2 \approx 1.414213 \dots$
$4$	$4 = 2$	$3$	$3 \approx 1.732050 \dots$
$9$	$9 = 3$	$5$	$5 \approx 2.236067 \dots$
$16$	$16 = 4$	$10$	$10 \approx 3.162277 \dots$
$25$	$25 = 5$	$20$	$20 \approx 4.472135 \dots$

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 \cdot a \geq 0, for any real number a

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.

\frac{9}{1 6} = \frac{3 ^{2}}{4 ^{2}} \Rightarrow \frac{9}{1 6} = \frac{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 = \frac{3 6}{1 0 0} = \frac{6 ^{2}}{1 0 ^{2}} ⇓ \frac{3 6}{1 0 0} = \frac{6}{1 0} = 0.6

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 \cdot 2 \cdot 2 = 8 \Rightarrow 38 = 2

Cube roots and raising a number to the third power are operations that undo each other.

$3 a^{3} = a and (3 a)^{3} = a$

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

(- 2) (- 2) (- 2) = - 8 \Rightarrow 3 - 8 = - 2

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.

\frac{8}{1 2 5} = \frac{2 ^{3}}{5 ^{3}} \Rightarrow 3 \frac{8}{1 2 5} = \frac{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 = \frac{2 7}{1 0 0 0} = \frac{3 ^{3}}{1 0 ^{3}} ⇓ 3 \frac{2 7}{1 0 0 0} = \frac{3}{1 0} = 0.3

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.

2, 3, 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.

2 π = 1.41421356237 \dots = 3.14159265359 \dots

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 ∖ Q

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
$25$	$2 5^{2}$	Rational Number
$18$	$2 3^{2} \cdot 2$	Irrational Number
$1000$	$2 1 0^{3}$	Irrational Number
$327$	$3 3^{3}$	Rational Number
$325$	$3 5^{2}$	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 $2$ 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

2

2 .

2 is not a perfect square ⇓ 2 is an irrational number

A calculator can be used to find the exact value of

2 .

To write the square root sign

a,

push

2 ND

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

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 \dots \approx 1.4

Note that the value of

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.

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.

There are five sacks, each labeled with its weight in kilograms using an irrational number in root form. The sacks contain different seeds.

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 $2 ND$ 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, $5 .$ Since it a square root, he needs to push $2 ND$ 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.

5 = 2.236067977 \dots \approx 2.24

Next, he will find the exact value of

35 .

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.

35 = 1.709975946 \dots \approx 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.

314 = 2.410142264 \dots \approx 2.41 20 = 4.472135955 \dots \approx 4.47 334 = 3.239611801 \dots \approx 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 ⇓ 35 < 5 < 314 < 334 < 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.

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. Their square roots are then calculated. The initial square root will be located between the two consecutive whole numbers. For example, consider

40

and several perfect squares close to

40 .

25, 36, 49, 64

Note that

40

is greater than

36

and less than

49 .

Therefore,

40

is between the square roots of these two values.

3636 < < 4040 < < 4949

Now, calculate the square roots of the two perfect squares.

366 < < 4040 < < 497

It can be concluded that the exact value of

40

is between

6

and

7 .

Next, the value will be rounded to one of these numbers. Which number is closer to

40 ?

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Catch-Up and Review

Extra

Extra

Hint

Solution