Non Perfect Squares, Square Roots, and Estimation
Textbook:
Page 44 – from CD, let’s do 1 2 3
Page 45 – from UA, do 1a
Page 49 – do “Not the Only Square Root”
Page 51 – do 8
Page 51 - Challenge do 12abc
13ab
1
Rational numbers have a decimal portion that
ends (finite)
3.45
or
repeats endlessly (infinitely)
87.19363636…
Irrational numbers have a decimal portion that never ends
and never repeats.
7.45932687…
The square root of a non-perfect square has a decimal
portion that never ends and never repeats. Therefore, the
square root of a non perfect square is an irrational
number.
6 is a non perfect square. Its square root never ends and
never repeats so its square root is an irrational number.
The exact answer for the square root of non perfect square
6 is √ 6
31 ?
47 ?
59 ?
2
Using your calculator, see that the approximate answer for
the √ 6 is 2.4494897…
non perfect square
exact square root
approx square root
2
3
5
6
7
8
11
13
3
27
29
31
others
1 4 9 16 25 36 49 64 81 100 … are examples of
perfect squares
1 2 3 4 5 6
7 8 9
10 are their square roots
2 3 5 6 7 8 10 11 13 14 15 17 18 19 20… are
non perfect squares
We can use a calculator to find their approximate square
roots and we can estimate to find their roots.
4
View the pattern for the estimation of the square root of 11 and then 13:
__
√11 is between the square roots of which two perfect squares?
__
√?
<
___
√9
<
3
<
__
√ 11
<
__
√?
___
√ 11
<
___
√ 16 =
___
√ 11
<
=
4
11 is closer to 9 than it is to 16. Therefore,
___
√ 11 is a little more than 3
___
The √ 13 is between the square roots of which two perfect squares?
__
___
__
√?
<
√ 13
<
√? =
___
√9
<
3
<
___
√ 13
<
___
√ 13
<
___
√ 16 =
4
13 is closer to 16 than it is to 9. Therefore,
__
√13 is a little less than 4
5
Now, you practice the concept, by completing the following:
__
__
__
__
√ 26
√ 34
√ 41 √ 53
__
√ 75
__
√ 89
___
√ 92
____
√ 103
Follow the pattern:
___
___
The √ 9 is 3, the √900 is 30
___
____
The √ 16 is 4, the √1600 is 40
__
___
The √ 25 is 5, the √2500 is ?
__
The √ 36 is
___
the √3600 is ?
___
The √ 49 is
____
the √4900 is ?
___
The √ 64 is
____
the √6400 is ?
9 900
16 1600
25 2500
examples of Perfect Squares.
36 3600
49 4900
64 6400 are
6
11 1100 13 1300 are examples of Non Perfect Squares. We estimate
____
___
___
the √ 1100 and √1300 in the same manner as √11
__
√9
<
3
<
___
√ 11 <
___
√11 <
___
√ 16
4
___
√900 <
30 <
___
√ 1100
____
√ 1100
<
____
√ 1600
<
40
___
The √1100 is a little more than 30
Use the estimation method to find the approximate square root of the
following:
__
___
____
_____
____
√ 3300
√6500
√ 7700
√ 8200
√9500
Important:
Example: 8100 has an even set of zeros or its factors are 81 x 100 so
__
____
√ 81 x √ 100 =
9 x
10 =
90
Shortcut: Find the square root of the 81 and add one zero.
Challenge:
Does the method work for 810? How about 7900?
Quick: What is the square root of 100000000?
7