Download MA.8.A.6.3 Simplify real number expressions using the laws of

MA.8.A.6.3 Simplify real number expressions using the laws of exponents.
DEFINITIONS
The product of repeated factors can be expressed as a power. A power consists of a base and an exponent. The exponent
tells how many times the base is used as a factor.
EXAMPLES
Example 1 Write each expression using exponents.
a. 7 · 7 · 7 · 7
7 · 7 · 7 · 7 = 74
The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent.
b. y · y · x · y · x
y·y·x·y·x=y·y·y·x·x
= (y · y · y) · (x · x)
= y3 · x2
Commutative Property
Associative Property
Definition of exponents
To evaluate a power, perform the repeated multiplication to find the product.
Example 2 Evaluate (-6)4.
(-6)4 = (-6) · (-6) · (-6) · (-6)
= 1,296
Write the power as a product.
Multiply.
The order of operations states that exponents are evaluated before multiplication, division, addition, and
subtraction.
Example 3 Evaluate m2 + (n − m)3 if m = -3 and n = 2.
m2 + (n − m)3 = (-3)2 + (2 − (-3))3
Replace m with -3 and n with 2.
= (-3)2 + (5)3
Perform operations inside parentheses.
= (-3 · -3) + (5 · 5 · 5)
Write the powers as products.
= 9 + 125 or 134
Add.
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Example 4 Evaluate m3 • m5
m3 = m • m • m
m5 = m • m • m • m • m
m • m • m • m • m • m • m • m = m8
m 3 • m 5 = m8
Notice the exponents of m3 • m5 add up to 8
So, when multiplying common bases, simply add the
exponents.
m5
m3
Example 5 Evaluate
mmmmm
mmm
mmmmm
mmm
5
m
= m2
3
m
Example 6 Evaluate
Write out each term in expanded notation.
Write out each term in expanded notation.
Cancel out as many m’s as possible.
Notice the exponents of m5 • m3 subtract to make 2
So, when multiplying common bases, simply subtract the
exponents.
m3
m5
mmm
mmmmm
mmm
mmmmm
3
m
1
= 2 = m–2
5
m
m
Write out each term in expanded notation.
Cancel out as many m’s as possible.
Notice the exponents of m3 • m5 subtract to make –2
So, when multiplying common bases, simply subtract the
exponents.
Example 7 Evaluate (m3 )2
(m3 )2 = m3 • m3
= m • m • m • m • m• m
m • m • m • m • m • m = m6
(m3 )2 = m6
Write out each term in expanded notation.
Notice the exponents of (m3 )2 multiply to 6
So, when raising a power to a power, simply multiply the
exponents.
Example 8: Simplify 5a3b-2
2
To do this, you need to know some more properties of exponents:
upstairs
. Negative exponents merely need to be moved. If it is upstairs, move it down. If it is
downstairs
5a 3b 2
downstairs, move it up. Turn 5a3b-2 into a fraction by putting it over 1.
1
Think of a fraction as

-2
Then, move the b downstairs change the negative 2 to positive 2.

4a 3b 0c 2
Example 9: Simplify
2a 2c
Look at each part separately.
5a 3
b2

First the numbers:

Then the a:
4 2

2 1
a3
move the a-2 upstairs
a 2
a5
1
 since anything to the 0 power is 1, b0 equals 1
Then the b:

Then the c:

c 2
c 2
since c doesn’t have an exponent, put a 1 for the exponent: 1
c
c
1
now move the c-2 downstairs: 3
c

Put it all together:
2a 5
c3


Example 10: When you have a fraction to the negative power, simply use the reciprocal of the fraction to the positive
power.
3 2 4 2 4 2  
16
       2 
4  3  3  9

or
a 2b 3 5 b 6 5 a 3 5 a15
 3    3    6   30
ab  a  b  b

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MA.8.A.6.3
Practice
Problems
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8.
9.
10.
11.
12. 32 · 33 = 3?
13. Find the value of the expression 43 - 33.
14. Numa loves beads and wants to know which amount
would be more, a thousand beads or (62)3 beads?
15. The teacher marked Silvano's problem wrong on his
test.
(45)4 = 49
Explain what he did wrong and give the correct answer.
16. Dmitry calculated that he needs 6s2 square inches of
wood for each crate he makes. Simplify the expression
when s is replaced by t4.
17. What is the area of the square:
5
18. The area of the rectangle in the figure is 24a2b3 square
units. Find the width of the rectangle.
19. A publisher sells 106 copies of a new book. Each book
has 102 pages. How many pages total are there in all of the
books sold? Write the answer using exponents.
20. Randall has 23 pairs of rabbits on his farm. Each pair of
rabbits can be expected to produce 25 baby rabbits in a year.
How many baby rabbits will there be on Randall's farm
each year? Write the answer using exponents.
21. A company has set aside 107 dollars for annual
employee bonuses. If the company has 104 employees and
the money is divided equally among them, how much will
each employee receive?
22. After making a down payment, Mr. Valle will make 62 monthly payments of 63 dollars each to pay for his new car.
What is the total of the monthly payments?
23.
24.
25.
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