Transcript Exponent Rules (PowerPoint)

Exponent Rules
Dr. Sarah Ledford
Mathematics Educator
[email protected]
Prior Standards
 In 5th grade, students use whole number exponents to
denote powers of 10.
 They also write numbers in expanded form such
345.678 as (3 x 100) + (4 x 10) + (5 x 1) +
(6 x 1/10) + (7 x 1/100) + (8 x 1/1000).
 In 6th grade, students work with numerical expressions
involving whole number exponents.
Exponents
82 = 8 x 8 = 64
In the calculator: 8 ^ 2
Exponents
32 = 3 x 3
23 = 2 x 2 x 2
34 = 3 x 3 x 3 x 3
43 = 4 x 4 x 4
56 = 5 x 5 x 5 x 5 x 5 x 5
63 = 6 x 6 x 6
In your own words?
Exponents
The Calculator is
always Correct?
My class was asked to simplify the expression
5 + 42 – 6 ÷ 3.
James’ calculator showed the answer to be 10.
Alexis’ calculator showed 19.
Lucy didn’t use a calculator but got 5.
Jorge also didn’t use a calculator and got 25.
Who is correct?
From GADOE 6th grade unit 3
The Calculator is
always Correct?
The correct solution is 19.
This problem shows the importance of having an order of
operations because without them, we can get several
different answers.
A scientific calculator can apply the order of operations
where a regular four-function calculator can not.
Equivalent?
Determine which of the two expressions are equivalent:
22  32 – 23 – 1
22  (32 – 23)– 1
(2  3)2 – 23 – 1
From GADOE 6th grade unit 3
Exponent Experimentation 2
https://www.illustrativemathematics.org/contentstandards/6/EE/A/1/tasks/2224
Here are some different ways to write the value 16:
24
12 – (21 + 22) + 500 ÷ 50
2/3 x 481 – (1 + 3)2
23 + 23
Exponent Experimentation 2
Find at least three different ways to write
each value below.
Include at least one exponent in all of
the expressions you write.
81
25
64/9
Exponent of 1
You can use your calculator to compute these if needed:
21 = ??
31 = ??
51 = ??
81 = ??
a1 = ??
If the exponent tells us how many of the base we have to
multiply, then an exponent of 1 tells us we have only 1 of the
base number.
In your own words?
Product Rule
What does each statement mean?
23 x 24 = ?
(2 x 2 x 2) x 24 = ?
(2 x 2 x 2) x (2 x 2 x 2 x 2) = ?
Are the parentheses necessary?
2 x 2 x 2 x 2x 2 x 2 x 2 = ?
How many 2’s are we multiplying together?
27 = ?
Product Rule
What does each statement mean?
32 x 34 = ?
(3 x 3) x 34 = ?
(3 x 3) x (3 x 3 x 3 x 3) = ?
Are the parentheses necessary?
3 x 3 x 3 x 3 x 3 x 3= ?
How many 3’s are we multiplying together?
36 = ?
Product Rule
What does each statement mean?
41 x 42 = ?
4 x 42 = ?
4 x (4 x 4) = ?
Are the parentheses necessary?
4x4x4=?
How many 4’s are we multiplying together?
43 = ?
Product Rule
23 x 24 = 27
32 x 34 = 36
41 x 42 = 43
Observations?
Can you come up with a rule that will always work?
am x an = a(m + n)
In your own words?
Quotient Rule
What does each statement mean?
Quotient Rule
What does each statement mean?
2x2x2x2=?
24 = ?
Quotient Rule
What does each statement mean?
Quotient Rule
What does each statement mean?
4x4=?
42 = ?
Quotient Rule
What does each statement mean?
Quotient Rule
What does each statement mean?
3=?
31 = ?
Quotient Rule
Observations?
Can you come up with a rule that will always work?
Quotient Rule
Observations?
Can you come up with a rule that will always work?
In your own words?
Power Rule
What does each statement mean?
(23)4 = ?
(2 x 2 x 2)4 = ?
(2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) = ?
Are the parentheses necessary?
2 x 2 x 2 x 2x 2 x 2 x 2 x 2 x 2 x 2x 2 x 2 = ?
How many 2’s are we multiplying together?
212 = ?
Power Rule
What does each statement mean?
(32)3 = ?
(3 x 3)3 = ?
(3 x 3) x (3 x 3) x (3 x 3) = ?
Are the parentheses necessary?
3x3x3x3x3x3=?
How many 3’s are we multiplying together?
36 = ?
Power Rule
What does each statement mean?
(54)2 = ?
(5 x 5 x 5 x 5)2 = ?
(5 x 5 x 5 x 5) x (5 x 5 x 5 x 5) = ?
Are the parentheses necessary?
5x5x5x5x5x5x5x5=?
How many 5’s are we multiplying together?
58 = ?
Power Rule
(23)4 = 212
(32)3 = 36
(54)2 = 58
Observations?
Can you come up with a rule that will always work?
(am)n = a(m x n)
In your own words?
Expanded Power Rule
What does each statement mean?
(3z)2 = ?
3z x 3z = ?
(3 x z) x (3 x z) = ?
(3 x 3) x (z x z) = ?
32 x z2 = ?
32 z2 = ?
9 z2
Expanded Power Rule
What does each statement mean?
(2de)3 = ?
2de x 2de x 2de = ?
(2 x d x e) x (2 x d x e) x (2 x d x e) = ?
(2 x 2 x 2) x (d x d x d) x (e x e x e) = ?
23 x d3 x e3 = ?
23 d3 e3 = ?
8 d3 e3
Expanded Power Rule
What does each statement mean?
Expanded Power Rule
What does each statement mean?
Expanded Power Rule
What does each statement mean?
Expanded Power Rule
What does each statement mean?
Expanded Power Rule
What does each statement mean?
Expanded Power Rule
(3z)2 = 32 z2
(2de)3 = 23 d3 e3
Observations?
Can you come up with rules that always work?
(ab)m = am bm
Expanded Power Rule
(3z)2 = 32 z2
(2de)3 = 23 d3 e3
Observations?
Can you come up with rules that always work?
(ab)m = am bm
In your own words?
Exploring Powers of 10
Adapted from GADOE GSE Framework 8th Grade Unit 2
On your paper going down in a column, write
106, 105, 104, …, 100, 10-1, 10-2, 10-3.
Simplify each. You may use a calculator if needed.
If answer is given as a decimal, convert it to a fraction.
Observations?
Observations?
 If power is positive, the answers are whole numbers.
 If power is positive, the bigger the power, the bigger the
answer.
 If power is negative, the answers are decimals or fractions.
 If power is positive, the power tells how many zeroes are
in the answer.
 If you look at the answers going down the page, they are
getting smaller.
 If you look at the answers going down the page, they are
getting smaller by a factor of 10.
Observations?
 If you look at the answers going up the page, they are
getting bigger.
 If you look at the answers going up the page, they are
getting bigger by a factor of 10.
 Is 100 weird? It fits the pattern…
 If power is negative, the power tells how many decimal
places are in the answer.
 If power is negative, the denominator is same as the
answer for when the power is positive. Hmmm…
Exploring Powers of 2
Let’s see if those observations hold for powers of 2.
On your paper going down in a column, write
26, 25, 24, …, 20, 2-1, 2-2, 2-3.
Simplify each. You may use a calculator if needed.
If answer is given as a decimal, convert it to a fraction.
Observations?
Observations?
 If power is positive, the answers are whole numbers.
 If power is positive, the bigger the power, the bigger the
answer.
 If power is negative, the answers are decimals or fractions.
 If you look at the answers going down the page, they are
getting smaller.
 If you look at the answers going down the page, they are
getting smaller by a factor of 2.
Observations?
 If you look at the answers going up the page, they are
getting bigger.
 If you look at the answers going up the page, they are
getting bigger by a factor of 2.
 Is 20 weird? It fits the pattern…
 If power is negative, the power tells how many decimal
places are in the answer. Does this hold for powers of 2
like it did for powers of 10?
 If power is negative, the denominator is same as the
answer for when the power is positive. Hmmm…
Negative Exponents
Observations?
Can you come up with a rule that will always work?
Negative Exponents
Observations?
Can you come up with a rule that will always work?
In your own words?
Zero Exponent Rule
100 = 1
20 = 1
50 = ?
990 = ?
100000 = ?
0.0050 = ?
You may use a calculator.
Zero Exponent Rule
100 = 1
50 = 1
20 = 1
990 = 1
100000 = 1
0.0050 = 1
Observations?
Can you come up with a rule that will always work?
a0 = 1
In your own words?
Raising to the Zero and
Negative Powers
https://www.illustrativemathematics.org/contentstandards/8/EE/A/1/tasks/1438
In this problem c represents a positive number.
The quotient rule for exponents says that if m and n are
positive integers with m > n, then
Raising to the Zero and
Negative Powers
A. What expression does the quotient
rule provide for
when m = n?
Raising to the Zero and
Negative Powers
Raising to the Zero and
Negative Powers
B. If m = n, simplify
without using the quotient rule.
Raising to the Zero and
Negative Powers
Anything divided by itself is 1.
Raising to the Zero and
Negative Powers
C. What do the previous two questions
suggest is a good definition for c0?
c0 = 1
(for any positive number c)
Raising to the Zero and
Negative Powers
D. What expression does the quotient
rule provide for
Raising to the Zero and
Negative Powers
Raising to the Zero and
Negative Powers
E. What expression do we get for
if we use the value for c0
found in part C?
Raising to the Zero and
Negative Powers
Raising to the Zero and
Negative Powers
F. Using parts D & E, propose
a definition for the expression
c – n.
Raising to the Zero and
Negative Powers
Exponent Rules
Mathematics Assessment Project
 http://map.mathshell.org
 Tools for formative and summative assessment that
make knowledge and reasoning visible, and help
teachers to guide students in how to improve, and
monitor their progress.
 Formative Assessment Lessons (FALs)  grades 6-11
MAP FALs
Lesson Plans – Concept development vs.
Problem Solving
Process – think independently & jot down
ideas/work, work within a group to come up
with a better/more efficient solution, discuss
student solutions, & discuss provided student
solutions
PPT slides
Worksheets
Applying Properties of
Exponents
http://map.mathshell.org/download.php?fileid=1668
Pre & post assessments
Card sort
Powers of 2
Write these as powers of 2.
A:
8 × 4 = 32
B:
16 ÷ 8 = 2
C:
8 ÷ 16 = ½
D:
8÷ 8 = 1
P-63
Card Sort
There are S cards and E cards.
Select a card and find all other cards that have the same
value as the one you have chosen.
Most only have an S card and E card match.
Some match to more than one E card or S card.
*On a sheet of paper, you can write down the simplified
solutions to help you match them.
Card Sort
Solution
S1 & E8
S6 & S10
& E10 & E12
S2 & E9
S7 & E5 & E14
S3 & E13
S8 & E1 & E7
S4 & E4
S9 &E6
S5 & E2
E3 & E11