Exponents 25 = ?

Vocabulary exponent – the number of times a number is multiplied by itself. base – the number that is being multiplied. exponent 83 base This is read “8 to the 3rd power” or “8 cubed.” It means 8 x 8 x 8.

Evaluating Exponents 25 = 2 x 2 x 2 x 2 x 2 = 32 = 6 x 6 x 6 = 216 63 1.34 = 1.3 x 1.3 x 1.3 x 1.3 = 2.8561

Exponents with a base of 10 Any multiple of ten can be expressed as an exponent with a base of ten. The base is 10. The number of zeroes gives us the exponent. Example: 100 = 102 10,000 = 104 1,000,000 = 106

Writing in Expanded Form Using Powers of 10 First, write the problem in expanded form. Then, change each term to a multiplication of the value and its place. Change the place values to exponents with powers of 10.

Example 7, 946 7, 000 + 900 + 40 + 6 (7 x 1,000) + (9 x 100) + (4 x 10) + (6 x 1) 7 x 103 + 9 x 102 + 4 x 101 + 6

Simplify. 1. 2  2  2 2. 3  3  3  3 3. 5  5  5 4. 4  4  4 5. 6  6  6  6  6 8 81 125 64 7,776

m • m • m = m3 Powers and Exponents A simple way to think of exponents and powers is as repeated multiplication m • m • m = m3

Key Terms exponent base factor: when two or more numbers are multiplied together, each number is called a factor exponent: an exponent tells how many times a number (called a base) is used as a factor power: a number that is expressed using a base and an exponent exponent base

Powers and Exponents

Evaluating Powers Find each value. A. 44 44 = 4  4  4  4 Use 4 as a factor 4 times. = 256 B. 73 73 = 7  7  7 Use 7 as a factor 3 times. = 343 C. 191 191 = 19 Use 19 as a factor 1 time.

43 72 27 35 106 171 Evaluating exponents Use 4 as a factor 3 times 4  4  4 = 16 43 72 27 35 106 171 Use 7 as a factor 2 times 7  7 = 49 Use 2 as a factor 7 times 2  2  2  2  2  2  2 = 128 Use 3 as a factor 5 times 3  3  3  3  3 = 243 Use 10 as a factor 6 times 10  10  10  10  10  10 = 1,000,000 Use 17 as a factor 1 time 171 = 17

Exponents To express a whole number as a power, write the number as the product of equal factors. Then write the product using the base and an exponent. For example, 10,000 = 10  10  10  10 = 104.

Expressing Whole Numbers as Powers Write each number using an exponent and the given base. A. 625, base 5 625 = 5  5  5  5 5 is used as a factor 4 times. = 54 B. 64, base 2 64 = 2  2  2  2  2  2 2 is used as a factor 6 times. = 26

Write each number using an exponent and the given base 7  7  7 = 73 256, base 2 2 2 2 2 2 2 2 2 = 28 1,000,000,000 base 10 10 10 10 10 10 10 10 10 10 = 109

Application On Monday, Erik tells 3 people a secret. The next day each of them tells 3 more people. If this pattern continues, how many people besides Erik will know the secret on Friday? On Monday, 3 people know the secret. On Tuesday, 3 times as many people know as those who knew on Monday. On Wednesday, 3 times as many people know as those who knew on Tuesday. On Thursday, 3 times as many people know as those who knew on Wednesday.

Application: Continued On Friday, 3 times as many people know as those who knew on Thursday. Each day the number of people is 3 times greater. 3  3  3  3  3 = = On Friday 243 people besides Erik will know the secret. 35 243

Scientists and mathematicians use exponential notation To describe very large and very small numbers. The average distance between the earth and the sun is 1 AU (astronomical unit) or about 93 million miles. We can write that number as Ninety three million 93,000,000 93 x 106 What advantage is there in using exponential notation to write very large numbers?