Math 6H ( Periods 3, 6, & 7)

Square Numbers & Square Roots (Continued) 5-3

Before we began---We reviewed the terminology for various numbers using 75 as our example
75---the standard form of the number
3⋅ 5⋅ 5 is it written in expanded prime factorization
3⋅ 5² is it's exponential prime factorization

If you have a prime number all three are exactly the same
For example
41 --- is the standard form of the number
but 41 is also the expanded prime factorization
and 41 ( or 41¹) is it written in exponential prime factorization.
[You can leave off the exponent 1 for the power of 1 because it is truly invisible!!]


Knowing the SQ's & SQRT's of numbers up to 20 is critical for finding the approximate square root of any non perfect square.
That is,
You know the √36 = 6
and √25 = 5
But what would be
√28 ?

There are several strategies to use.. one involved dividing and taking the average. This is from the yellow worksheet handed out on Friday 12/4
You know that √28 is less than 6 but more than 5
so you start with 5
divide 28 by 5
28 ÷ 5 = 5.6
Now take the average of 5 and 5.6 or
5 + 5.6
2

which equals 5.3

Divide 28 by 5.3 now
28 ÷ 5.3 = equals 5.28 or rounded 5.3
Since the divisor (5.3) and the quotient (5.3) match STOP...and say

√28≈ 5.3

That's fairly complicated... in class we showed some neat tricks about the relationship between the perfect squares...
Using the same number √28

You still think of the perfect square below and above

Stack them
36
√28
25

then take the square roots for those perfect squares

√36 = 6
√28 ≈
√25 = 5

√36 = 6
√28 ≈ 5
√25 = 5

Now, find the difference between your two perfect squares
36 -25 = 11
We discovered that the difference will always be the sum of the square roots of the perfect squares!! WOW!!

That number becomes your denominator

Now, find the difference between your number (in this case 28) and the lower of the two perfect squares (25) so 28-25 = 3
That number becomes the numerator
so you have

3/11

Now, we haven't learned about fractions but you can estimate ( since this is all about estimating... anyway)
and you know

3/10
That can be written as .3
so adding that to your estimate
we can safely estimate
√28 ≈ 5.3

Try finding the √110... with this method..