Download Standard M8N1: Students will understand different representations

Standard
M8N1: Students will understand different representations of numbers including square
roots, exponents, and scientific notation.
g. Simplify, add, subtract, multiply, and divide expressions containing
square roots
Squares and Square Roots
Definitions:
Square Roots - the number multiplied by itself to find a perfect square
Perfect Square - the product that is the result of multiplying square
roots
Radicals - expressions involving square root symbols
Radicand - the number inside the square root symbol (√)
The following square roots and their perfect squares should be memorized:
Square Perfect
Root Square
Square Perfect
Root Square
1
1
13
169
2
4
14
196
3
9
15
225
4
16
16
256
5
25
17
189
6
36
18
324
7
49
19
361
8
64
20
400
9
81
25
625
10
100
30
900
11
121
35
1225
12
144
40
1600
45
2025
50
2500
When a radicand is not a perfect square, estimate the square root.
√30 = 5
5 x 5 = 25
and
6 x 6 = 36
25 is closest to 30, so the best estimate
for the square root of 30 is 5
All perfect squares have two square roots:
a negative and a positive square root.
√25 = -5 and 5
-5 x -5 = 25;
WHY?
Negative x Negative = Positive
and 5 x 5 = 25
Definitions:
Integers - The set of whole numbers and their opposites.
Rational Numbers - All numbers that can be expressed as the ratio of two
integers,
, with m ≠ 0. This includes any positive fraction, negative fraction,
positive or negative mixed numbers.
Irrational Numbers - Numbers that cannot be expressed as a ratio of two integers.
This means that these numbers in decimal form are nonterminating and
nonrepeating.
A common example of an irrational number is to find the square root of
a number that is not a perfect square.
Example: √5 = 2.2360697.......
Simplifying Square Roots
If the radicand in a square root radical is not a perfect square, you can simply find the
greatest perfect-square factor of the radicand.
√72 Find the greatest perfect-square
factor of 72
72 = 9 x 8
9 is a perfect square - its square root is 3; 8 is not a perfect
square
4 x 18
4 is a perfect square - its square root is 2; 18 is not a perfect
sq.
36 x 2 are also factors of 72 - Type equation here.36 is a perfect square
but 2 is not
Therefore, 72 has three factors that are perfect squares
Since 36 is the largest perfect square of 72, we will write the √72 as √36 x 2. We
know that the square root of 36 is 6. We will pull it out of the radical term. It will now
look like this: √72 = 6√2
Example:
Which of the radicands is a perfect square? 16 is the perfect square; its square root is
4. We can rewrite the radical term as
Factors of 80 = 16 x 5
The greatest perfect square factor of 80
is 16. We take the perfect square out of
80 and now we have
The greatest perfect square factor of 80 is 16. We take the perfect square out of 80
and now we have
Cancel the common numerator and denominator
Adding and Subtracting Square Roots
If two radical expressions have the same radicand, you can add and subtract the
expressions using the distributive property.
Add: 6√5 + 8√5 = √5(6 + 8)
14√5
7√8 + 3√8 = √8 (7 + 3)
10√8
Practice:
5√3 - 2√3 =
√24 + √28 + √54 =
√75 + 2√12 =
Multiplying and Dividing
Square Roots
When you multiply radicals, you multiply the radicands.
Example:
6√3 x 5√15
Use the Commutative Property to reorder the factors. Then multiply:
6 x 5 x √3 x √15 =
30 x √3 x 15
30 x √45 = 30√9 x 5 =
30 x 3√5 = 90√5
Divide:
12√32
4√2
Simplify the fraction:
12
√32
4
√2
3 x √16
3 x 4 = 12
Multiply:
Rationalizing the Denominator