7.4 Division Properties of Exponents 7.4 Division Properties of Exponents Algebra 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 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 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 ×10 9–5 = 0.58 ×10 4 = 5800

Check It Out! Example 3 In 1990, the United States public debt was about 3.2 × dollars. The population of the United States in 1990 was about 2.5 × 10 8 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 × dollars. The population of the United States in 1990 was about 2.5 × 10 8 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

Simplify. Additional Example 4A: Finding Positive Powers of Quotient Use the Power of a Quotient Property. 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: (x 3 y 3 ) 2 = x 32 y 32.

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.

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 (y 3 ) 2 = y 32 = y 6. Use the Power of a Product Property (2x 2 ) 2 = 2 2 x 22. 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 = 2 3 n 3 and (6m) 3 = 6 3 m 3.

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. 9 3 = 729 and 4 3 = 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: (b 2 c 3 ) 4 = b 24 c 34 = b 8 c 12 and (2a) 4 = 2 4 a 4 = 16a 4. 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 Simplify. 6. Simplify (3  ) ÷ (5  10 5 ) and write the answer in scientific notation. 6  The Republic of Botswana has an area of 6  10 5 square kilometers. Its population is about 1.62  What is the population density of Botswana? Write your answer in standard form. 2.7 people/km 2