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7.4 Division Properties of Exponents
Algebra 1Algebra 1
2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
California Standards
A quotient of powers with the same base can be found by writing the powers in factored form and dividing out common factors.
m factors
n factors
Simplify.
Additional Example 1: Finding Quotients of Powers
A. B.
C.
Simplify.
Additional Example 1: Finding Quotients of Powers
D.
Check It Out! Example 1
a.
Simplify.
b.
Check It Out! Example 1
Simplify.
c. d.
You can “split up” a quotient of products into a product of quotients:
Writing Math
Example:
Additional Example 2: Dividing Numbers in Scientific Notation
Simplify and write the
answer in scientific notation
Write as a product of quotients.
Simplify each quotient.Simplify the exponent.Write 0.5 in scientific notation
as 5 x 10 .The second two terms have the same
base, so add the exponents.
Simplify the exponent.
Check It Out! Example 2
Simplify and write the answer in scientific notation.
Write as a product of quotients.
Simplify each quotient.
Simplify the exponent.= 1.1 × 10–2
Additional Example 3: Economics Application
The Colorado Department of Education spent about dollars in fiscal year 2004-05 on public schools. There were about students enrolled in public school. What was the average spending per student? Write your answer in standard form.
To find the average spending per student, divide the dollars spent by the number of students.
Write as a product of quotients.
Additional Example 3 Continued
The Colorado Department of Education spent about dollars in fiscal year 2004-05 on public schools. There were about students enrolled in public school. What was the average spending per student? Write your answer in standard form.
The average spending per student is $5800.
Simplify each quotient.
Simplify the exponent.
Write in standard form.
= 0.58 ×109–5
= 0.58 ×104
= 5800
Check It Out! Example 3
In 1990, the United States public debt was about 3.2 × 1012 dollars. The population of the United States in 1990 was about 2.5 × 108 people. What was the average debt per person? Write your answer in standard form.
To find the average debt per person, divide the total debt by the number of people.
Write as a product of quotients.
Check It Out! Example 3 Continued
Simplify each quotient.
Simplify the exponent.
Write in standard form.
The average debt per person was $12,800.
In 1990, the United States public debt was about 3.2 × 1012 dollars. The population of the United States in 1990 was about 2.5 × 108 people. What was the average debt per person? Write your answer in standard form.
A power of a quotient can be found by first writing factors and then writing the numerator and denominator as powers.
n factors
n factors
n factors
Simplify.
Additional Example 4A: Finding Positive Powers of Quotient
Use the Power of a Quotient Property.
Simplify.
Simplify.
Additional Example 4B: Finding Positive Powers of Quotient
Use the Power of a Quotient Property.
Use the Power of a Product Property:
Simplify and use the Power of a Power Property:
Simplify.
Additional Example 4C: Finding Positive Powers of Quotient
Use the Power of a Quotient Property.
Use the Power of a Product Property:
Simplify.
Additional Example 4C Continued
Simplify.
Simplify and use the Power of a Power Property: .Use the Power of a Product Property: (x3y3)2 = x32y32.
Check It Out! Example 4a
Simplify.
Use the Power of a Quotient Property.
Simplify.
Check It Out! Example 4b Simplify.
Check It Out! Example 4c Simplify.
Simplify.
Additional Example 5A: Finding Negative Powers of Quotients
Rewrite with a positive exponent.
and
Use the Power of a Quotient Property .
Simplify.
Additional Example 5B: Finding Negative Powers of Quotients
Use the Power of a Quotient Property.
Use the Power of a Power Property (y3)2 = y32 = y6. Use the Power of a Product Property (2x2)2 = 22x22.
Simplify.
Simplify.
Additional Example 5C: Finding Negative Powers of Quotients
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.
1
241
21
12
Divide out common factors.
Simplify.
Additional Example 5C: Finding Negative Powers of Quotients
Simplify.
Whenever all of the factors in the numerator or the denominator divide out, replace them with 1.
Helpful Hint
Check It Out! Example 5a
Simplify.
93 = 729 and 43 = 64.
Use the Power of a Quotient Property.
Rewrite with a positive exponent.
Check It Out! Example 5b
Simplify.
Rewrite with a positive exponent.
Use the Power of a Product Property: (b2c3)4= b2•4c3•4 = b8c12 and (2a)4= 24a4 = 16a4.
Use the Power of a Quotient Property.
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.
Lesson Quiz 7.4
1.
3. 4.
5.
2.
Simplify.
6. Simplify (3 1012) ÷ (5 105) and write the answer in scientific notation. 6 106
7. The Republic of Botswana has an area of 6 105 square kilometers. Its population is about 1.62 106. What is the population density of Botswana? Write your answer in standard form.
2.7 people/km2
