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**Division Properties of Exponents**

7-4 Division Properties of Exponents Cornell Notes Due: Thursday, February 19, 2015 Holt Algebra 1

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Objective How do you 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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**Both and 729 are considered to be simplified.**

Helpful Hint

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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**

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 The second two terms have the same base, so add the exponents. Simplify the exponent.

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**You can “split up” a quotient of products into a product of quotients:**

Example: Writing Math

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Check It Out! Example 3 In 1990, the United States public debt was about dollars. The population of the United States was about 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.

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**Check It Out! Example 3 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 per person? Write your answer in standard form. 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 Product Property. Use the Power of a Product Property: Simplify and use the Power of a Power Property:

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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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**. 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,

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**Example 5A: Finding Negative Powers of Quotients**

Simplify. Rewrite with a positive exponent. Use the Powers of a Quotient Property . and

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**Example 5B: Finding Negative Powers of Quotients**

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: (3)2 (2n)3 = 32 23n3 and (2)2 (6m)3 = 22 63m3

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**Example 5C: Finding Negative Powers of Quotients**

Simplify. Square and cube terms. 1 24 2 12 Divide out common factors. Simplify.

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

Helpful Hint

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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 Power Property: (b2c3)4= b2•4c3•4 = b8c12 and (2a)4= 24a4= 16a4.

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