Transcript Diapositiva 1 - St. Landry Parish School Board

Squares, Square Roots, Cube
Roots, & Rational vs.
Irrational Numbers
Perfect Squares
• Can be represented by arranging objects in a square.
Perfect Squares
Perfect Squares
• 1x1=1
• 2x2=4
• 3x3=9
• 4 x 4 = 16
Activity: Calculate the perfect squares
up to 152…
•
•
•
•
•
•
•
•
1x1=1
2x2=4
3x3=9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
8 x 8 = 64
Perfect Squares
 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
Square Numbers
4cm
4cm
16 cm2
• 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.
Square Numbers
• The large square has an area
of 4cm x 4cm = 16 cm2.
4cm
4cm
16 cm2
• The number 4 is called the
square root of 16.
• We write: 4 =
16
The opposite of squaring a number is taking
the square root.
81
This 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
9 X 9 = 81
Is there another solution
to this problem?
Yes!!!
-9 X -9 = 81
as well!
So… 9 & -9
are square
roots of 81
Simplify Each Square Root
100
10
 16
-4
Simplify Each Square Root
64
8
 49
-7
What About Fractions?
Take the square root of
numerator and the square root
of the denominator
1
9
=
1
3
1
3
What About Fractions?
So…the square root of
1
9
is…………
1
3
What About Fractions?
Take the square root of
numerator and the square root
of the denominator
9
100
=
3
10
3
10
What About Fractions?
So…the square root of
9
100
is…………
3
10
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
Estimate the value of each
Example:
expression to the nearest integer.
28
Is not a perfect square but it
does fall between two perfect
squares.
25 and 36
Estimating Square Roots
25
28
36
5
6
Since 28 is closer to 25 than it is to 36,
28 ≈ 5
Estimating Square Roots
Estimate the value of each
Example:
expression to the nearest integer.
45
Is not a perfect square but it
does fall between two perfect
squares.
36 and 49
Estimating Square Roots
36
45
6
49
7
Since 45 is closer to 49 than it is to 36,
45 ≈ 7
Estimating Square Roots
Estimate the value of each
Example:
expression to the nearest integer.
 105
Is not a perfect square but it
does fall between two perfect
squares.
-100 and -121
Estimating Square Roots
 100  105
-10
 121
-11
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. Irrational
Real 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. Irrational
Examples:
- 0.81
5
2
1
9
Rational; the decimal repeats
Irrational; not a perfect square
Rational; is a fraction
0.767667666... Irrational; decimal does not
terminate or repeat
π
Rational vs. Irrational
Practice:
7
64
- 0.456
Irrational; Pi is a decimal that
does not terminate or repeat
Irrational; not a perfect square
Rational; is a perfect square
Rational; the decimal terminates
Cube Roots
To “Cube” a number we multiply it
by itself three times
4
3
4
=
3
=
4 x 4x4
64
Cube Roots
Remember that taking the “cube
root” of a number is the opposite
of cubing a number.
3
125
=
5 x 5x5
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.
 27
3
=
-3 x -3 x -3
- 3 is the cube root of - 27
Simply Each Cube Root
3
1000
10
 216
-6
3
Simply Each Cube Root
3
729
 8
3
9
-2