Explore Square Roots and Cube Roots: Understanding Irrational Numbers, Radicands and Indices

	$n$ 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.

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.

a < 0

No real roots.

One negative root, which is the principal root.

Principal Root of Perfect Squares	Principal Root of Non-Perfect Squares
Perfect Square	Principal Root (Integer Number)	Non-Perfect Square	Principal Root (Irrational Number)
$1$	$1 = 1$	$2$	$2 \approx 1.414213 \dots$
$4$	$4 = 2$	$3$	$3 \approx 1.732050 \dots$
$9$	$9 = 3$	$5$	$5 \approx 2.236067 \dots$
$16$	$16 = 4$	$10$	$10 \approx 3.162277 \dots$
$25$	$25 = 5$	$20$	$20 \approx 4.472135 \dots$

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

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

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

a Recall that a square as a geometric figure is a rectangle with four equal side lengths. The area of a square can be found by finding the square of a side length.

Side Length a Area a^{2}

The numbers written in the centers of the cultivated fields are the squares of their corresponding side lengths. An inverse operation can be applied to find the side lengths. To undo the operation of squaring, calculate the square root of each area.

Area a^{2} Side Length a^{2} = a

Note that

9

is a perfect square because it can be written as the square of

3 .

With this in mind, now take the square root of

9

to find the side length of the given field.

Area 9 Side Length 3^{2} = 3

The side length of the corn field is

3

yards.

b Apply the same process from Part A to calculate the side length of the tomato field. Take square root of

36

to undo the operation of squaring. Since

36

is the square of

6,

it is a perfect square.

Area a^{2} 36 Side Length a^{2} = a 6^{2} = 6

So, the side length of the tomato field is

6

yards.

c The side length of the last field, the clover field, will be calculated now. Note that

121

is a perfect square because it is the square of

11 .

To find the side length of the field, take the square root of

121 .

Area a^{2} 121 Side Length a^{2} = a 1 1^{2} = 11

The side length of the clover field is

11

yards.

d This time, it is asked to find the common characteristics between all the squares. The side lengths will be first examined in a table to check whether they are integers, multiples of

3,

irrational numbers, or rational numbers.

	Integer	Multiple of $3$	Irrational Number	Rational Number
$3$	$✓$	$✓$	$\times$	$✓$
$6$	$✓$	$✓$	$\times$	$✓$
$11$	$✓$	$\times$	$\times$	$✓$

According to the table, all the side lengths are integers and rational numbers, but not all are multiples of

3

or irrational numbers. As a final option, consider whether all the areas are perfect squares.

Area 936121 Perfect Square ? 9 = 3^{2} ✓ 36 = 6^{2} ✓ 121 = 1 1^{2} ✓

All the areas are perfect squares because all of them are the square of an integer. It can be concluded that three of the given options are common characteristics between all the squares.

All the side lengths are integers.

All the side lengths are rational numbers.

All the areas are perfect squares.

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Catch-Up and Review

Finding the Side Lengths of Squares

$n^{th}$ Root

Principal Root

Finding Index and Radicand

Commonly Used Roots

Square Root

Extra

Cube Root

Extra

Evaluating Roots

Irrational Numbers

Irrational Number or Not?

Is $2$ an Irrational Number?

Finding the Amounts of Seeds

Hint

Solution

Rounding Square Roots

Rounding Cube Roots

Estimating Roots

Finding the Side Lengths of Squares

Hint

Solution

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