squares, square roots, cube roots, & rational vs. irrational numbers

Squares, Square Roots, Cube Roots, & Rational vs. Irrational Numbers

Perfect Squares• Can be represented by arranging objects in a square.

Perfect Squares

• 1 x 1 = 1

• 2 x 2 = 4

• 3 x 3 = 9

• 4 x 4 = 16

Activity: Calculate the perfect squares up to 152…

Perfect Squares

• 1 x 1 = 1

• 2 x 2 = 4

• 3 x 3 = 9

• 4 x 4 = 16

• 5 x 5 = 25

• 6 x 6 = 36

• 7 x 7 = 49

• 8 x 8 = 64

9 x 9 = 81 10 x 10 =

100 11 x 11 =

121 12 x 12 =

144 13 x 13 =

169 14 x 14 =

196 15 x 15 =

225

Perfect Squares

Square Numbers• One property of a perfect

square is that it can be represented by a square

array.

• Each small square in the array shown has a side

length of 1cm.

• The large square has a side length of 4 cm.

4cm

4cm 16 cm2

• The large square has an area of 4cm x 4cm = 16 cm2.

• The number 4 is called the square root of 16.

• We write: 4 = 16

4cm

4cm 16 cm2

Square Numbers

The opposite of squaring a number is taking the square root.

81This is read “the square root of 81” and is asking “what number can be multiplied by itself and equal 81?”

9 X 9 = 81 so The square root of 81 is 9

81 Is there another solution

to this problem?

9 X 9 = 81

Yes!!!So… 9 & -9 are square roots of 81

-9 X -9 = 81 as well!

100

16

Simplify Each Square Root

10

- 4

Simplify Each Square Root

8

- 7

64

49

What About Fractions?

=19

1

3

1

3

Take the square root of numerator and the square root

of the denominator

What About Fractions?19

1

3

So…the square root of

is…………

What About Fractions?

=9

100

3

10

3

10

Take the square root of numerator and the square root

of the denominator

What About Fractions?9

1003

10

So…the square root of

is…………

Think About It

Do you see that squares and square roots are

inverses (opposites) of each other?

Estimating Square Roots

Not all square roots will end-up with perfect whole numbers

When this happens, we use the two closest perfect squares that the number falls between and get

an estimate

Estimating Square Roots

Example: Estimate the value of each expression to the nearest integer.

28Is not a perfect square but it does fall between two perfect squares.

25 and 36

Estimating Square Roots

25 36

5 6

28

Since 28 is closer to 25 than it is to 36,

28 ≈ 5

Estimating Square Roots

Example: Estimate the value of each expression to the nearest integer.

45Is not a perfect square but it does fall between two perfect squares.

36 and 49

Estimating Square Roots

36 49

6 7

45

Since 45 is closer to 49 than it is to 36,

45 ≈ 7

Estimating Square Roots

Example: Estimate the value of each expression to the nearest integer.

105Is not a perfect square but it does fall between two perfect squares.

-100 and -121

Estimating Square Roots

100 121

-10 -11

105

Since -105 is closer to -100 than it is to -121,

105 ≈ -10

Estimating Square Roots

Practice: Estimate the value of the expression to the nearest integer.

22 ≈ - 5

54 ≈ 7

Rational vs. IrrationalReal Numbers – include all rational and

irrational numbers

Rational Numbers – include all integers, fractions, repeating, terminating decimals, and perfect squares

Irrational Numbers – include non-perfect square roots, non-terminating decimals, and non-repeating decimals

Rational vs. IrrationalExamples:

- 0.81 Rational; the decimal repeats

Irrational; not a perfect square

Rational; is a fraction

512

90.767667666... Irrational; decimal does not

terminate or repeat

Rational vs. IrrationalPractice:

Irrational; Pi is a decimal that does not terminate or repeat

Irrational; not a perfect square

Rational; is a perfect square

7

Rational; the decimal terminates- 0.456

64

π

Cube Roots To “Cube” a number we multiply it by itself three times

= 4 x 4 x 44 3

4 3 = 64

Cube Roots Remember that taking the “cube root” of a number is the opposite of cubing a number.

= 5 x 5 x 51253

5 is the cube root of 125

Cube Roots Remember that taking the “cube root” of a number is the opposite of cubing a number.

= -3 x -3 x -3 273

- 3 is the cube root of - 27

Simply Each Cube Root

10

- 6

10003

2163

Simply Each Cube Root

9

- 2

7293

83