Squares, Square Roots, Cube Roots,

1
Squares, Square Roots, Cube Roots, Rational vs.
Irrational Numbers
2
Perfect Squares

• Can be represented by arranging objects in a
  square.

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Perfect Squares
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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

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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

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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
16 cm2
4cm
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Square Numbers

• 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
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The opposite of squaring a number is taking the
square root.
This is read the square root of 81 and is
asking what number can be multiplied by itself
and equal 81?
The square root of 81 is 9
9 X 9 81
so
9
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
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Simplify Each Square Root
10
- 4
11
Simplify Each Square Root
8
- 7
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What About Fractions?
Take the square root of numerator and the square
root of the denominator

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What About Fractions?
Sothe square root of
is
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What About Fractions?
Take the square root of numerator and the square
root of the denominator

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What About Fractions?
Sothe square root of
is
16
Think About It
Do you see that squares and square roots are
inverses (opposites) of each other?
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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
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Estimating Square Roots
Estimate the value of each expression to the
nearest integer.
Example
Is not a perfect square but it does fall between
two perfect squares.
25 and 36
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Estimating Square Roots
5
6
Since 28 is closer to 25 than it is to 36,
5

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Estimating Square Roots
Estimate the value of each expression to the
nearest integer.
Example
Is not a perfect square but it does fall between
two perfect squares.
36 and 49
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Estimating Square Roots
6
7
Since 45 is closer to 49 than it is to 36,
7

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Estimating Square Roots
Estimate the value of each expression to the
nearest integer.
Example
Is not a perfect square but it does fall between
two perfect squares.
-100 and -121
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Estimating Square Roots
-10
-11
Since -105 is closer to -100 than it is to -121,
-10

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Estimating Square Roots
Practice Estimate the value of the
expression to the nearest integer.
- 5

7

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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
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Rational vs. Irrational
Examples
- 0.81
Rational the decimal repeats
Irrational not a perfect square
Rational is a fraction
Irrational decimal does not terminate or repeat
0.767667666...
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Rational vs. Irrational
Practice
p
Irrational Pi is a decimal that does not
terminate or repeat
Irrational not a perfect square
Rational is a perfect square
- 0.456
Rational the decimal terminates
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Cube Roots
To Cube a number we multiply it by itself three
times

4 x 4 x 4

64
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Cube Roots
Remember that taking the cube root of a number
is the opposite of cubing a number.

5 x 5 x 5
5 is the cube root of 125
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Cube Roots
Remember that taking the cube root of a number
is the opposite of cubing a number.

-3 x -3 x -3
- 3 is the cube root of - 27
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Simply Each Cube Root
10
- 6
32
Simply Each Cube Root
9
- 2