Transcript Exponents, Square Roots, Order of Operations

Squares & Square
Roots
Perfect Squares
Square Number
 Also called a “perfect square”
 A number that is the square of a
whole number
 Can be represented by
arranging objects in a square.
Square Numbers
Square Numbers
1x1=1
2x2=4
3x3=9
 4 x 4 = 16
Square Numbers
1x1=1
2x2=4
3x3=9
 4 x 4 = 16
Activity:
Calculate the perfect
squares up to 152…
Perfect Squares
1x1=1
 9 x 9 = 81
2x2=4
 10 x 10 = 100
3x3=9
 11 x 11 = 121
 4 x 4 = 16
 12 x 12 = 144
 5 x 5 = 25
 13 x 13 = 169
 6 x 6 = 36
 14 x 14 = 196
 7 x 7 = 49
 15 x 15 = 225
 8 x 8 = 64
Activity:
Identify the following numbers
as perfect squares or not.
i.
ii.
iii.
iv.
v.
vi.
16
15
146
300
64
121
Activity:
Identify the following numbers
as perfect squares or not.
i.
ii.
iii.
iv.
v.
vi.
16 = 4 x 4
15
146
300
64 = 8 x 8
121 = 11 x 11
Square Numbers
 One property of a perfect
4cm
4cm
16 cm2
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
Square Root
 A number which, when
multiplied by itself, results in
another number.
 Ex: 5 is the square root of 25.
5 =
25
Exponents
In the table below, the number 2 is written as a factor repeatedly. The product of factors is also displayed
in this table. Each product of factors is called a power.
Product of Factors
Factors
Description
Translation
2x2=
4
2 is a factor 2 times
4 is the 2nd power of 2
2x2x2=
8
2 is a factor 3 times
8 is the 3rd power of 2
2x2x2x2=
16
2 is a factor 4 times
16 is the 4th power of 2
2x2x2x2x2=
32
2 is a factor 5 times
32 is the 5th power of 2
2x2x2x2x2x2=
64
2 is a factor 6 times
64 is the 6 power of 2
2x2x2x2x2x2x2=
128
2 is a factor 7 times
128 is the 7th power of 2
2x2x2x2x2x2x2x2=
256
2 is a factor 8 times
256 is the 8th power of 2
Writing 2 as a factor one million times would be a very time-consuming and tedious task.
A better way to approach this is to use exponents.
Exponential notation is an easier way to write a number as a product of many factors.
BaseExponent
The exponent tells us how many times the base is used as a factor.
For example, to write 2 as a factor one million times, the base is 2, and the exponent is 1,000,000.
We write this number in exponential form as follows:
21,000,000
2n
read as two raised to the millionth power
read as two raised to the nth power.
Example 1:
Write 2 x 2 x 2 x 2 x 2 using exponents, then read your answer
aloud.
Solution:
2 x 2 x 2 x 2 x 2 = 25
“2 raised to the fifth power”
Let us take another look at the table from above to see how exponents work.
Factor
Form
Standard
Form
22 =
2x2=
4
23 =
2x2x2=
8
24 =
2x2x2x2=
16
25 =
2x2x2x2x2=
32
26 =
2x2x2x2x2x2=
64
27 =
2x2x2x2x2x2x2=
128
28 =
2x2x2x2x2x2x2x2=
256
Exponential
Form
So far we have only examined numbers with a base of 2. Let's look at some examples of writing exponents where the
base is a number other than 2.
Example 2:
Write 3 x 3 x 3 x 3 using exponents, then read your answer aloud.
Solution:
3 x 3 x 3 x 3 = 34
Example 3:
Write 6 x 6 x 6 x 6 x 6 using exponents, then read your answer aloud.
Solution:
6 x 6 x 6 x 6 x 6 = 65
Example 4:
Write 8 x 8 x 8 x 8 x 8 x 8 x 8 using exponents, then read your answer aloud.
Solution:
8 x 8 x 8 x 8 x 8 x 8 x 8 = 87
Example 5:
Solution:
3 raised to the fourth power
6 raised to the fifth power
8 raised to the seventh power
Write 103, 36, and 18 in factor form and in standard form.
Exponential
Form
Factor
Form
103
10 x 10 x 10
36
3x3x3x3x3x3
18
1x1x1x1x1x1x1x1
Standard
Form
1,000
729
1
Notice in powers of
10, the exponent
tells you how many
zeroes come after
the 1.
102 = 100
103 = 1,000
104 = 10,000
So far we have only examined numbers with a base of 2. Let's look at some examples of writing exponents where the
base is a number other than 2.
Expand each notation and find it’s value
a) 23
P. 94 EXAMPLE 3
This means to re-write the expression as a
product of repeated factors. The
exponent tells you how many times to
repeat the factor (which is the base)
Solution:
p.95 EX 6
2x2x2=8
Write 102, 103, and 104 in factor form and in standard form.
Exponential
Form
Factor
Form
102
10 x 10
103
10 x 10 x 10
104
10 x 10 x 10x 10
Another wording for “exponential form” is to use the word
“power”.
100 is the 2nd power of 10, or 102
Exampe 7 b) Write 10,000,000 as a power of 10.
How many zeroes? ___
107
Standard
Form
100
1,000
10,000
Notice in powers of
10, the exponent
tells you how many
zeroes come after
the 1.
102 = 100
103 = 1,000
104 = 10,000
Special Exponents
The following rules apply to numbers with exponents of 0, 1, 2 and 3:
Example
Rule
Any number (except 0) raised to the zero power is equal to 1.
1490 = 1
Any number raised to the first power is always equal to itself.
81 = 8
If a number is raised to the second power, we say it is squared.
32 is read as three squared
If a number is raised to the third power, we say it is cubed.
43 is read as four cubed
Why does any nonzero number to the zero power equal 1?
It just makes sense in the pattern of exponents.
Each exponent that is one less than the previous one is the power divided by the base.
24 = 16
23 = 8
22 =
÷2
÷2
34 =
33 =
4
÷2
32 =
21= 2
÷2
31=
20= 1
30=
P
E
MD
AS
Order of Operatons
Please: do all operations within parentheses and other grouping symbols (such as [ ], or operations in
numerators and denominators of fractions) from innermost outward.
Excuse: calculate exponents
My,Dear: do all multiplications and divisions as they occur from left to right
Aunt,Sally: do all additions and subtractions as they occur from left to right.
Example:
20 – 2 + 3(8 - 6)2 Expression in parentheses gets calculated first
= 20 – 2 + 3(2)2 Next comes all items with exponents. The exponent only applies to the item directly to the left
of it. In this case, only the (2) is squared.
= 20 – 2 + 3(4) Next in order comes multiplication. Multiplication and Division always come before addition or
division, even if to the right.
= 20 – 2 + 12 Now when choosing between when to do addition and when to do subtraction, always go from
left to right, so do 20-2 first, because the subtraction is to the left of the addition.
= 18 + 12 Now finally we can do the addition.
= 30
Order of Operations with
Square Roots
Make sure you follow PEMDAS
Treat the Radical Sign as a special grouping symbol.
Example :
 6 81  5 1
Do what’s inside the “grouping symbols” first.
= - 6(9) + 5(1)
= -54 + 5
= -49