Use division properties of exponents to evaluate and simplify expressions'

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Objective
Use division properties of exponents to evaluate
and simplify expressions.
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A quotient of powers with the same base can be
found by writing the powers in a factored form
and dividing out common factors.
Notice the relationship between the exponents in
the original quotient and the exponent in the
final answer 5 3 2.
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Example 1 Finding Quotients of Powers
Simplify.
A.
B.
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Example 1 Finding Quotients of Powers
Simplify.
C.
D.
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Check It Out! Example 1
Simplify.
a.
b.
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Check It Out! Example 1
Simplify.
c.
d.
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Example 2 Dividing Numbers in Scientific Notation
Write as a product of quotients.
Simplify each quotient.
Simplify the exponent.
The second two terms have the same base, so add
the exponents.
Simplify the exponent.
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Check It Out! Example 2
Write as a product of quotients.
Simplify each quotient.
Simplify the exponent.
The second two terms have the same base, so add
the exponents.
Simplify the exponent.
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Example 3 Application
To find the average spending per student, divide
the total debt by the number of students.
Write as a product of quotients.
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Example 3 Continued
To find the average spending per student, divide
the total debt by the number of students.
0.58 1095
Simplify each quotient.
Simplify the exponent.
0.58 104
Write in standard form.
5800
The average spending per student is 5800.
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Check It Out! Example 3
To find the average debt per person, divide the
total debt by the number of people.
Write as a product of quotients.
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Check It Out! Example 3 Continued
To find the average debt per person, divide the
total debt by the number of people.
Simplify each quotient.
Simplify the exponent.
Write in standard form.
The average debt per person was 12,800.
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A power of a quotient can be found by first
writing the numerator and denominator as powers.
Notice that the exponents in the final answer are
the same as the exponent in the original
expression.
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Example 4A Finding Positive Powers of Quotient
Simplify.
Use the Power of a Quotient Property.
Simplify.
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Example 4B Finding Positive Powers of Quotient
Simplify.
Use the Power of a Quotient Property.
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Example 4C Finding Positive Powers of Quotient
Simplify.
Use the Power of a Quotient Property.
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Example 4C Continued
Simplify.
Simplify.
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Check It Out! Example 4a
Simplify.
Use the Power of a Quotient Property.
Simplify.
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Check It Out! Example 4b
Simplify.
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Check It Out! Example 4c
Simplify.
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Write the fraction as division.
Use the Power of a Quotient Property.
Multiply by the reciprocal.
Simplify.
Use the Power of a Quotient Property.
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Example 5A Finding Negative Powers of Quotients
Simplify.
Rewrite with a positive exponent.
Use the Power of a Quotient Property .
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Example 5B Finding Negative Powers of Quotients
Simplify.
Use the Power of a Quotient Property.
Use the Power of a Power Property (y3)2 y3?2
y6. Use the Power of a Product Property (2x2)2
22x2?2
Simplify.
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Example 5C Finding Negative Powers of Quotients
Simplify.
Rewrite each fraction with a positive exponent.
Use the Power of a Quotient Property.
Use the Power of a Product Property (2n)3 23n3
and (6m)3 63m3.
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Example 5C Finding Negative Powers of Quotients
Simplify.
Divide out common factors.
Simplify.
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Check It Out! Example 5a
Simplify.
Rewrite with a positive exponent.
Use the Power of a Quotient Property.
93 729 and 43 64.
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Check It Out! Example 5b
Simplify.
Rewrite with a positive exponent.
Use the Power of a Quotient Property.
Use the Power of a Product Property (b2c3)4
b24c34 b8c12 and (2a)4 24a4 16a4.
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Check It Out! Example 5c
Simplify.
Rewrite each fraction with a positive exponent.
Use the Power of a Quotient Property.
Simplify.
Add exponents and divide out common terms.