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Name Class Date Practice
Form G
Division Properties of Exponents
Simplify each expression.
56
1. 2 625
5
5
55
2. 2
5
1
x8
3. 3 x 4
4.
x8
6.
4
3
7.a 5 b 81
3
49
16
12p2
11. °
¢
15p
13. a
-4
m2
625
3x2y 5z -2 -3 125z 21
b 5xz 5
27x 3y 15
7m 2
1
3m4
10. a-
2
256p
1
21m4
3x
8. a 2y b
625
-2
m-3
m-5
3
x6 y 9
5. 2 5 x 4y 4
x y
9.a 4
7b
125
3
3x4
b
2y 5
ab3
12. a 5 b
a b
14.
27x 3
8y 3
-3
8y 15
27x 12
-2 a8
b4
(4m2) (3n5)
−6m2n2
(2m-3) ( - mn)3
Explain why each expression is not in simplest form.
15. 24 r 3 16. (3x)2
24 is not simplified
both factors should be squared
y5
18. y
17. m3 n0 n0 = 1, so m3n0 = m3
y5
4
y can be simplified to y
Simplify each quotient. Write each answer in scientific notation.
19.
3.6 * 107
2.4 × 104
1.5 * 103
20.
4.5 * 10-6
9.0 × 10−5
5 * 10-2
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Name Class Date Practice (continued)
Form G
Division Properties of Exponents
21. Writing Explain how you divide expressions with numerators and
denominators written in scientific notation. How do you handle the
exponents? What do you do with the coefficients? Connect your response to
the rules you have learned regarding the division properties of exponents.
For expressions with numerators and denominators in scientific notation, you
work separately with the numerical parts and the powers of 10. To divide powers
of 10, subtract the exponents. Put the answer back into scientific notation form.
22. A computer can do a computation in 6.8 * 10-9 seconds. How many
computations can the computer do in 5 minutes? 4.41 × 1010
3
64
23. Error Analysis A student simplifies the expression a 2 b as follows:
3
3
64
4-2
3
2
3
a 2 b = [(6 , 3) ] = (2 ) = 64. What mistake did the student make in
3
simplifying the expression? What is the correct simplification?
The student should have expanded the powers inside the brackets first to follow
( # ## # )
6 6 3
= (2
the order of operations. 6 6
3 3
# 2 # 6 # 6)3 = 1443 = 2,985,984
24. Reasoning The division property of exponents says that to simplify powers
with the same base you subtract the exponents. Use examples to show why
powers need to have the same base in order for this technique to work.
Answers may vary. Sample:
65
=6
63
65
=6
33
# 6 # 6 # 6 # 6 = 6 # 6 = 36: 65 = 65−3 = 62 = 36; same
6 # 6 # 6
63
# 6 # 6 # 6 # 6 = 6 # 6 # 2 # 2 # 2 = 288: 65 = ( 6 )2 = 22 = 4; different
3 # 3 # 3
3
33
same base = same answer; different bases = different answers; The division
property works only with numbers with the same bases.
25. The area of a triangle is 80x 5 y 3 . The height of the triangle is x4 y. What is the
length of the base of the triangle? 160xy 2
3
12m5
26. Open-Ended First simplify the expression a 15m b by raising each factor in
the parentheses to the third power and next reducing the result. Then simplify
by some other method. Explain your method. Are the results the same? Which
method do you prefer?
5 3
( 12m
15m )
=
15
12
(12m5)3
;
= 1728m 3 = 64m
125
(15m)3
3375m
5 3
( 12m
15m )
3
( )
4
= 4m
5
12
Simplifying
= 64m
125
within the parantheses first and then raising each factor to the thrid power gives
the same results. I prefer this method because the numbers are easier to calculate.
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