A quotient of powers with the same base

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.

Example 1: Finding Quotients of Powers Simplify. A. B.

Example 1: Finding Quotients of Powers Simplify. C. D.

Helpful Hint Both and 729 are considered to be simplified.

Check It Out! Example 1 Simplify. a. b.

Check It Out! Example 1 Simplify. c. d.

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.

Writing Math You can “split up” a quotient of products into a product of quotients: Example:

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.

Example 3: 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 scientific notation. To find the average spending per student, divide the total debt by the number of students. Write as a product of quotients.

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 scientific notation. To find the average spending per student, divide the total debt by the number of students. Simplify each quotient. Simplify the exponent.

Check It Out! Example 4 In 1990, the United States public debt was about dollars. The population of the United States was about people. What was the average debt person? Write your answer in scientific notation. To find the average debt person, divide the total debt by the number of people. Write as a product of quotients.

Check It Out! Example 4 Continued In 1990, the United States public debt was about dollars. The population of the United States was about people. What was the average debt person? Write your answer in scientific notation. To find the average debt person, divide the total debt by the number of people. Simplify each quotient. Simplify the exponent.

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.

Example 4 A: Finding Positive Powers of Quotient Simplify. Use the Power of a Quotient Property. Simplify.

Example 4 B: Finding Positive Powers of Quotient Simplify. Use the Power of a Product Property: Simplify and use the Power of a Power Property:

Example 4 C: Finding Positive Powers of Quotient Simplify. Use the Power of a Product Property:

Example 4 C Continued Simplify. Use the Power of a Product Property:

Check It Out! Example 4 a Simplify. Use the Power of a Quotient Property. Simplify.

Simplify. Check It Out! Example 4 b

Simplify. Check It Out! Example 4 c

Remember that . What if x is a fraction? Write the fraction as division. Use the Power of a Quotient Property. Multiply by the reciprocal. Simplify. Use the Power of a Quotient Property. Therefore,

Example 5 A: Finding Negative Powers of Quotients Simplify. Rewrite with a positive exponent. Use the Powers of a Quotient Property. and

Example 5 B: Finding Negative Powers of Quotients Simplify.

Example 5 C: 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: (3)2 (2 n)3 = 32 23 n 3 and (2)2 (6 m)3 = 22 63 m 3

Example 5 C: Finding Negative Powers of Quotients Simplify. Square and cube terms. 1 1 1 2 24 Divide out common factors. 12 Simplify.

Helpful Hint Whenever all of the factors in the numerator or the denominator divide out, replace them with 1.

Check It Out! Example 5 a Simplify. Rewrite with a positive exponent. Use the power of a Quotient Property. 93=729 and 43 = 64.

Check It Out! Example 5 b Simplify. Rewrite with a positive exponent. Use the Power of a Quotient Property. Use the Power of a Power Property: (b 2 c 3)4= b 2 • 4 c 3 • 4 = b 8 c 12 and (2 a)4= 24 a 4= 16 a 4.

Simplify. Check It Out! Example 5 c Rewrite each fraction with a positive exponent. Use the Power of a Quotient Property. Use the Power of a Product Property: (3)2= 9. Add exponents and divide out common terms.

Lesson Quiz: Part I Simplify. 1. 2. 3. 4. 5. 6. Simplify (3 1012) ÷ (5 105) and write the answer in scientific notation. 6 106