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

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

Integer

Multiple of

3

Irrational Number

Rational Number

3

✓

✓

\times

✓

6

✓

✓

\times

✓

11

✓

\times

\times

✓

The following figure is made up of five identical squares.

The area of each square is

36

square centimeters. What is the perimeter of this figure?

We are given that the area of each square is 36 square centimeters. We want to find the perimeter of the figure, which means that we need to find the side length of one of the squares. Remember that we can find the area of a square by finding the square of one side length. cc Side Length & Area a & a^2 Since we know the area of a square, we can find one side length of a square by taking the square root of the area. cc Area & Side Length a^2 & sqrt(a^2)=a Now let's calculate the square root of 36. Notice that 36 is a perfect square.

The side length of one square is 6 centimeters. Since the squares are identical, that means that the side length of all of the squares is 6 centimeters! Let's see all the side lengths on the figure!

As shown, there are twelve equal sides of the figure in total. Now we will add them up, or we can multiply 12 times 6 centimeters to find the perimeter of the figure. 12 * 6 = 72 The perimeter of the figure is 72 centimeters.

Determine when the following sentence is true.

If a certain number is a perfect square and also a perfect cube, then its square root and cube root are the same number.

We know that a certain number is both a perfect square and a perfect cube. We will determine whether it is possible that its square root and its cube root be the same number and how often this happens. Let's recall the definition of a perfect square first.

Perfect Square |- A perfect square is a number that is the square of an integer.

Now let's remember the definition of a perfect cube.

Perfect Cube |- A perfect cube is a number that is a cube of an integer.

Let's list the perfect squares and the perfect cubes that are between 1 and 100.

Perfect Squares	Perfect Cubes
1, 4, 9, 16, 25, 36, 49, 64, 81, 100	1, 8, 27, 64

We can see that the numbers 1 and 64 are both perfect squares and perfect cubes. 1 &= 1 * 1 = 1 * 1 * 1 64 &= 8 * 8 = 4 * 4 * 4 Let's calculate the square roots and cube roots of these numbers. c Square Root & Cube Root [0.3em] sqrt(1) = 1 & sqrt(1) = 1 [0.3em] sqrt(64) = 8 & sqrt(64) = 4 We can see that for 1, the square root and the cube root are the same. However, for 64, the square root and cube root are different. Therefore, the square root and cube root of a number that is a perfect square and perfect cube at the same time are sometimes the same number.

Roots

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

Roots

Recommended exercises

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