Algebra 1 7 3 Multiplication Properties of Exponents

Algebra 1 7. 3 Multiplication Properties of Exponents

California Standards 2. 0 Students understand use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand use the rules of exponents.

You have seen that exponential expressions are useful when writing very small or very large numbers. To perform operations on these numbers, you can use properties of exponents. You can also use these properties to simplify your answer. In this lesson, you will learn some properties that will help you simplify exponential expressions containing multiplication.

Products of powers with the same base can be found by writing each power as a repeated multiplication. am an = (a a … a) m factors n factors = a a … a = am+n m + n factors

Additional Example 1: Finding Products of Powers Simplify. A. Since the powers have the same base, keep the base and add the exponents. B. Group powers with the same base together. Add the exponents of powers with the same base.

Additional Example 1: Finding Products of Powers Simplify. C. Group powers with the same base together. Add the exponents of powers with the same base. D. Group the first two powers. n 0 1 The first two powers have the same base, so add the exponents. Add the exponents.

Remember! A number or variable written without an exponent actually has an exponent of 1. 10 = 101 y = y 1

Check It Out! Example 1 Simplify. a. Since the powers have the same base, keep the base and add the exponents. b. Group powers with the same base together. Add the exponents of powers with the same base.

Check It Out! Example 1 Simplify. c. Group powers with the same base together. Add.

Check It Out! Example 1 Simplify. d. Group powers with the same base together. Divide the first group and add the second group. Multiply.

Additional Example 2: Astronomy Application Light from the Sun travels at about miles per second. It takes about 15, 000 seconds for the light to reach Neptune. Find the approximate distance from the Sun to Neptune. Write your answer in scientific notation. distance = rate time Write 15, 000 in scientific notation. Use the Commutative and Associative Properties to group. Multiply within each mi group. Neptune is about 2. 79 x 109 miles from the Sun.

Check It Out! Example 2 Light travels at about 1. 86 × 105 miles per second. Find the approximate distance that light travels in one hour. Write your answer in scientific notation. distance = rate time Write 3, 600 in scientific notation. Use the Commutative and Associative Properties to group. Multiply within each group. 8 Light will travel 6. 696 × 10 miles in one hour.

To find a power of a power, you can use the meaning of exponents. = a m am … am n factors = a a … a a … a = amn m factors n groups of m factors

Additional Example 3: Finding Powers of Powers Simplify. Use the Power of a Power Property. Zero multiplied by any number is zero. 1 Any number raised to the zero power is 1.

Additional Example 3: Finding Powers of Powers Simplify. C. Use the Power of a Power Property. Simplify the exponent of the first term. Since the powers have the same base, add the exponents. Write with a positive exponent.

Check It Out! Example 3 Simplify. Use the Power of a Power Property. Simplify. Use the Power of a Power Property. Zero multiplied by any number is zero. 1 Any number raised to the zero power is 1.

Check It Out! Example 3 c Simplify. c. Use the Power of a Power Property. Simplify the exponents of the two terms. Since the powers have the same base, add the exponents.

Powers of products can be found by using the meaning of an exponent. (ab)n = ab … ab n factors = a a … a b b … b = a n bn n factors

Additional Example 4: Finding Powers of Products Simplify. A. Use the Power of a Product Property. Simplify. B. Use the Power of a Product Property. Simplify.

Caution! In Example 4 A, the negative sign is not part of the base. –(2 y)2 = – 1 (2 y)2

Additional Example 4: Finding Powers of Products Simplify. C. Use the Power of a Product Property. Use the Power of a Power Property. Simplify.

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

Check It Out! Example 4 Simplify. c. Use the Power of a Product Property. Use the Power of a Power Property. Simplify. Write with a positive exponent.

Simplify. Lesson Quiz 7. 3 1. 32 • 34 2. 3. (x 3)2 4. 5. 6. 7. 8. The islands of Samoa have an approximate area of 2. 9 103 square kilometers. The area of Texas is about 2. 3 102 times as great as that of the islands. What is the approximate area of Texas? Write your answer in scientific notation. 6. 67 × 105 km 2