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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
