1 5 Properties of of Exponents Warm Up

1 -5 Properties of of Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra 22

1 -5 Properties of Exponents Warm Up Simplify. 1. 4 4 4 64 2. 3. 4. 20 5. 7. 3 104 30, 000 Holt Algebra 2 6. 105 100, 000

1 -5 Properties of Exponents Objectives Simplify expressions involving exponents. Use scientific notation. Holt Algebra 2

1 -5 Properties of Exponents Vocabulary scientific notation Holt Algebra 2

1 -5 Properties of Exponents In an expression of the form an, a is the base, n is the exponent, and the quantity an is called a power. The exponent indicates the number of times that the base is used as a factor. Holt Algebra 2

1 -5 Properties of Exponents When the base includes more than one symbol, it is written in parentheses. Reading Math A power includes a base and an exponent. The expression 23 is a power of 2. It is read “ 2 to the third power” or “ 2 cubed. ” Holt Algebra 2

1 -5 Properties of Exponents Example 1 A: Writing Exponential Expressions in Expanded Form Write the expression in expanded form. (5 z)2 (5 z) Holt Algebra 2 The base is 5 z, and the exponent is 2. 5 z is a factor 2 times.

1 -5 Properties of Exponents Example 1 B: Writing Exponential Expressions in Expanded Form Write the expression in expanded form. –s 4 –(s s s s) = –s s Holt Algebra 2 The base is s, and the exponent is 4. s is a factor 4 times.

1 -5 Properties of Exponents Example 1 C: Writing Exponential Expressions in Expanded Form Write the expression in expanded form. 3 h 3(k + 3)2 3(h)(h)(h) (k + 3) Holt Algebra 2 There are two bases: h and k + 3. h is a factor 3 times, and k + 3 is a factor 2 times.

1 -5 Properties of Exponents Check It Out! Example 1 a Write the expression in expanded form. (2 a)5 (2 a)(2 a)(2 a) Holt Algebra 2 The base is 2 a, and the exponent is 5. 2 a is a factor 5 times.

1 -5 Properties of Exponents Check It Out! Example 1 b Write the expression in expanded form. 3 b 4 3 b b Holt Algebra 2 The base is b, and the exponent is 4. b is a factor 4 times.

1 -5 Properties of Exponents Check It Out! Example 1 c Write the expression in expanded form. –(2 x – 1)3 y 2 There are two bases: 2 x– 1, and y. –(2 x – 1)(2 x – 1) y y Holt Algebra 2 2 x– 1 is a factor 3 times, and y is a factor 2 times.

1 -5 Properties of Exponents Holt Algebra 2

1 -5 Properties of Exponents Caution! Do not confuse a negative exponent with a negative expression. Holt Algebra 2

1 -5 Properties of Exponents Example 2 A: Simplifying Expressions with Negative Exponents Simplify the expression. 3– 2 The reciprocal of Holt Algebra 2 .

1 -5 Properties of Exponents Example 2 B: Simplifying Expressions with Negative Exponents Simplify the expression. The reciprocal of Holt Algebra 2 .

1 -5 Properties of Exponents Check It Out! Example 2 a Simplify the expression. 32 3 3=9 Holt Algebra 2 The reciprocal of .

1 -5 Properties of Exponents Check It Out! Example 2 b Write the expression in expanded form. (– 5)– 5 æ 1ö ç- ÷ è 5ø Holt Algebra 2 5 The reciprocal of .

1 -5 Properties of Exponents Holt Algebra 2

1 -5 Properties of Exponents Example 3 A: Using Properties of Exponents to Simplify Expressions Simplify the expression. Assume all variables are nonzero. 3 z 7(– 4 z 2) 3 (– 4) z 7 z 2 – 12 z 7 + 2 – 12 z 9 Holt Algebra 2 Product of Powers Simplify.

1 -5 Properties of Exponents Example 3 B: Using Properties of Exponents to Simplify Expressions Simplify the expression. Assume all variables are nonzero. (yz 3 – 5)3 = (yz– 2)3 Quotient of Powers y 3(z– 2)3 Power of a Product y 3 z(– 2)(3) Power of a Product Negative of Exponent Property Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 3 a Simplify the expression. Assume all variables are nonzero. (5 x 6)3 53(x 6)3 Power of a Product 125 x(6)(3) Power of a Power 125 x 18 Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 3 b Simplify the expression. Assume all variables are nonzero. (– 2 a 3 b)– 3 Negative Exponent Property Power of a Power Holt Algebra 2

1 -5 Properties of Exponents Scientific notation is a method of writing numbers by using powers of 10. In scientific notation, a number takes a form m 10 n, where 1 ≤ m <10 and n is an integer. You can use the properties of exponents to calculate with numbers expressed in scientific notation. Holt Algebra 2

1 -5 Properties of Exponents Example 4 A: Simplifying Expressions Involving Scientific Notation Simplify the expression. Write the answer in scientific notation. 3. 0 10– 11 Holt Algebra 2 Divide 4. 5 by 1. 5 and subtract exponents: – 5 – 6 = – 11.

1 -5 Properties of Exponents Example 4 B: Simplifying Expressions Involving Scientific Notation Simplify the expression. Write the answer in scientific notation. (2. 6 104)(8. 5 107) (2. 6)(8. 5) (104)(107) 22. 1 1011 Multiply 2. 6 and 8. 5 and add exponents: 4 + 7 = 11. 2. 21 1012 Because 22. 1 > 10, move the decimal point left 1 place and add 1 to the exponent. Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 4 a Simplify the expression. Write the answer in scientific notation. 0. 25 10– 3 2. 5 Holt Algebra 2 10– 4 Divide 2. 325 by 9. 3 and subtract exponents: 6 – 9 = – 3. Because 0. 25 < 10, move the decimal point right 1 place and subtract 1 from the exponent.

1 -5 Properties of Exponents Check It Out! Example 4 b Simplify the expression. Write the answer in scientific notation. (4 10– 6)(3. 1 10– 4) (4)(3. 1) (10– 6)(10– 4) 12. 4 10– 10 1. 24 10– 9 Holt Algebra 2 Multiply 4 by 3. 1 and add exponents: – 6 – 4 = – 10. Because 12. 4 >10, move the decimal point left 1 place and add 1 to the exponent.

1 -5 Properties of Exponents Example 5: Problem-Solving Application Light travels through space at a speed of about 3 105 kilometers per second. Pluto is approximately 5. 9 1012 m from the Sun. How many minutes, on average, does it take light to travel from the Sun to Pluto? Holt Algebra 2

1 -5 Properties of Exponents Example 5 Continued 1 Understand the Problem The answer will be the time it takes for light to travel from the Sun to Pluto. List the important information: • The speed of light in space is about 3 105 kilometers per second. • The distance from the Sun to Pluto is 5. 9 1012 meters. Holt Algebra 2

1 -5 Properties of Exponents Example 5 Continued 2 Make a Plan Use the relationship: rate, or speed, equals distance divided by time. Holt Algebra 2

1 -5 Properties of Exponents Example 5 Continued 3 Solve First, convert the speed of light from There are 1000, or 103 meters in every kilometers and 60 seconds in every minute Holt Algebra 2

1 -5 Properties of Exponents Example 5 Continued 3 Solve Use the relationship between time, distance, and speed to find the number of minutes it takes light to travel from the Sun to Pluto. It takes light approximately 328 minutes to travel from the Sun to Pluto. Holt Algebra 2

1 -5 Properties of Exponents Example 5 Continued 4 Look Back Light travels at 3 105 km/s for 328(60) ≈ 19, 666 seconds travels a distance of 5, 899, 560, 000 = 5. 89 109 km or 5. 89 1012 m. The answer is reasonable. Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 5 Light travels through space at a speed of about 3 105 kilometers per second. Earth is approximately 1. 5 1011 m from the Sun. How many minutes, on average, does it take light to travel from the Sun to Earth? Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 5 Continued 1 Understand the Problem The answer will be the time it takes for light to travel from the Sun to Earth. List the important information: • The speed of light is about 3 105 kilometers per second. • The distance from the Sun to Earth is 1. 5 1011 meters. Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 5 Continued 2 Make a Plan Use the relationship: rate, or speed, equals distance divided by time. Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 5 Continued 3 Solve First, convert the speed of light from There are 1000, or 103 meters in every kilometers and 60 seconds in every minute Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 5 Continued 3 Solve Use the relationship between time, distance, and speed to find the number of minutes it takes light to travel from the Sun to Earth. It takes light approximately 8. 333 minutes to travel from the Sun to Earth. Holt Algebra 2

1 -5 Properties of Exponents Check It Out! Example 5 Continued 4 Look Back Light travels at 1. 49 108 km or 1. 49 1011 m for 8. 33(60) ≈ 499. 99 seconds travels a distance of 149, 999, 400 = 3 105 kilometers per second. The answer is reasonable. Holt Algebra 2

1 -5 Properties of Exponents Lesson Quiz: Part I Simplify each expression. 2. 1. 4– 3 Simplify each expression. Assume all variables are nonzero. 3. 8 y 6(– 6 y 3)– 48 y 9 4. Simplify each expression. Write the answer in scientific notation. 5. Holt Algebra 2 4. 0 10– 10

1 -5 Properties of Exponents Lesson Quiz: Part II Simplify each expression. Write the answer in scientific notation. 6. If light travels about 3 105 kilometers per second, about how many kilometers does it travel in 1 day? Write the answer in scientific notation. 2. 592 1010 km Holt Algebra 2