Lesson: 8.EE.A.1 Part 1
Part 1: Product Rule of Exponents
Without a calculator, expand the expression and find the value:
24 ∙ 23 =
32 ∙ 33 =
55 ∙ 54 =
What pattern do you notice?
Product Rule: for any nonzero rational number a and integers n and m, 𝑎𝑛 ∙ 𝑎𝑚 =
1.
Which expression is equivalent to 24 ∙ 24
a.
216
b.
28
c.
8
d.
2.
3.
4
4.
a.
Which expression is equivalent to
a.
(−5)8
b.
(−5)15
c.
(25)8
d.
(25)15
e.
(−25)15
∙
(−5)5
5.
3 2
3 3
5
5
3 4
4
4
3 6
(− )
4
9
8
b.
( )
c.
(− )
d.
(− )
416
(−5)3
3 2
Which expression is equivalent to (− ) ∙ (− )
16
9
6
16
9
8
16
Which expression is equivalent to 42 ∙ 43 ∙ 44
a.
6424
b.
649
c.
49
d.
424
Which expression is equivalent to ( ) ∙ ( )
a.
9
( )
6
25
9
b.
( )
c.
( )
d.
( )
6.
James wrote the following equation: 34 ∙ 9 = 36
Using properties of exponents, explain why he is
5
25
correct.
3 6
5
3 5
5
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Part II:
Without a calculator, expand the expression and find the value:
(23 )2 =
(33 )3 =
(25 )4 =
What pattern do you notice?
How is this pattern related to the rule from Part I?
Rule: for any nonzero rational number a and integers n and m, (𝑎𝑛 )𝑚 =
7. Which expression is equivalent to (43 )3
a. 46
b. 24
11. Select all expressions equivalent to (43 ∙
42 )4
c. 36
a. 412 ∙ 48
d. 49
b. 45 ∙ 44
c. 45 ∙ 45 ∙ 45 ∙ 45
8. What value of n would make the equation
d. (46 )4
true?
(2𝑛 )2 = 28
12. Select all expressions equivalent to ((−3)4 ∙
(−3)3 )2
a. (−3)9
9. What value of n would make the equation
b. (−3)14
true?
c. (−3)8 ∙ (−3)6
(43 ∙ 42 )𝑛 = 415
d. (−3)6 ∙ (−3)5
e. (−3)24
10. Which expression is equivalent to (2 ∙ 23 )2
a. 45
b. 46
c. 28
d. 26
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Part III: Quotient Rule of Exponents
In Part I, you discovered 24 ∙ 23 = 27
Because 24 ∙ 23 = 27 could be expanded to 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 27
Using your work in Part I and Part II, expand the following numerical expressions.
24 ÷ 23 =
36 ÷ 34 =
55 ÷ 52 =
What pattern do you notice?
Quotient Rule: for any nonzero rational numbers a and integers n and m, 𝑎𝑚 ÷ 𝑎𝑛 =
13. Which expression is equivalent to
34
?
31
a. 33
b.
4
1
c. 3−3
14. What value of n will make the equation
15. Select all expressions equivalent to
a.
23 ∙24
25
24
22
b. 22
c.
212
25
d.
47
25
true?
(45 ÷ 43 )𝑛 = 44
16. Which expression is equivalent to
33 ∙32
9
a. 1
b. 94
c. 33
d.
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3
9
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Part IV: Negative and Zero Exponents
In Part III above, you discovered the property of division of expressions with integer exponents. Use that work to
determine what happens when the exponent of the number in the denominator is greater than the exponent of the
number in the numerator:
Instead of using the property, we could rewrite it in the expanded form
53 ÷ 55 =
Thus, 5−2 =
25 ÷ 28 =
Thus, 2−3 =
What pattern do you notice?
Task:
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
𝑐𝑚
= 𝑐 𝑚−𝑛
𝑐𝑛
Using the quotient rule and the work you did above, complete the following exploration of the quotient rule
when 𝑚 ≤ 𝑛:
a.
What expression does the quotient rule provide for
𝑐𝑚
𝑐𝑛
𝑐𝑚
𝑐𝑛
when 𝑚 = 𝑛?
b.
c.
If 𝑚 = 𝑛, simplify
without using the quotient rule.
What do parts (a) and (b) above suggest is a good definition for 𝑐 0 ?
d.
What expression does the quotient rule provide for 𝑐 𝑛?
e.
f.
What expression do we get for 𝑛? if we use the value for 𝑐 0 found in part (c)?
𝑐
Using parts (d) and (e), propose a definition for the expression 𝑐 −𝑛 .
𝑐0
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𝑐0
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Negative Exponent Rule: For any nonzero rational number a and integer n, 𝑎−𝑛 =
Zero Exponent Rule: For any nonzero rational number a, 𝑎0 =
17. Select all expressions equivalent to (43 ∙ 42 )4
a.
412 ∙ 48
5
4
23. Select all expressions equivalent to
a.
2-5 · 2-1
b.
2-3 · 22
b.
4 ∙ 4
c.
45 ∙ 45 ∙ 45 ∙ 45
c.
2-2 · 2-4
d.
(46 )4
d.
21 · 25
e.
21 · 26
f.
22 · 2-8
g.
23 · 23
18. Select all expressions equivalent to
a.
(−3)9
b.
(−3)14
c.
(−3)8 ∙ (−3)6
d.
(−3)6 ∙ (−3)5
e.
(−3)24
19. Which expression is equivalent to
a.
b.
c.
((−3)4
∙
(−3)3 )2
24. Select all expressions equivalent to
34
31
?
3
b.
3−4
c.
32
33
e.
4
1
f.
3−3
20. What value of n will make the equation true?
3−8
3−4
−12
a.
d.
1
26
1
32
1
34
1
312
25. Show that
35
3−7
= 312
(45 ÷ 43 )𝑛 = 44
21. Select all expressions equivalent to
a.
b.
c.
d.
23 ∙24
25
26. Show that (2−2 )−3 = 26
24
22
22
27. What is 200?
212
25
47
28. What is 10?
25
22. Select all expressions equivalent to (45 ∙ 4−3 )−2
a.
1
256
b.
256
c.
4−10 ∙ 46
d.
43 ∙ 4−5
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