2-7 Properties of Exponents Warm Up Problem of the Day Lesson Presentation Pre-Algebra

2-7 Properties of Exponents Warm Up Evaluate. 1. 33 27 Pre-Algebra 2-7 Properties of Exponents Warm Up Evaluate. 1. 33 27 2. 4 • 4 • 4 • 4 256 3. b2 for b = 4 16 4. n2r for n = 3 and r = 2 18

Problem of the Day Calculate 6 to the fourth power minus 56.

Learn to apply the properties of exponents and to evaluate the zero exponent.

The factors of a power, such as 74, can be grouped in different ways The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product. 7 • 7 • 7 • 7 = 74 (7 • 7 • 7) • 7 = 73 • 71 = 74 (7 • 7) • (7 • 7) = 72 • 72 = 74

MULTIPLYING POWERS WITH THE SAME BASE Words Numbers Algebra To multiply powers with the same base, keep the base and add the exponents. bm • bn = bm + n 35 • 38 = 35 + 8 = 313 MULTIPLYING POWERS WITH THE SAME BASE

Additional Example 1A & 1B: Multiplying Powers with the Same Base Multiply. Write the product as one power. A. 66 • 63 6 6 + 3 Add exponents. 6 9 B. n5 • n7 n 5 + 7 Add exponents. n 12

Additional Example 1: Multiplying Powers with the Same Base Continued Multiply. Write the product as one power. C. 25 • 2 2 5 + 1 Think: 2 = 2 1 2 6 Add exponents. D. 244 • 244 24 4 + 4 Add exponents. 24 8

A. 42 • 44 4 Add exponents. 4 B. x2 • x3 x Add exponents. x Try This: Example 1A & 1B Multiply. Write the product as one power. A. 42 • 44 4 2 + 4 Add exponents. 4 6 B. x2 • x3 x 2 + 3 Add exponents. x 5

Cannot combine; the bases are not the same. Try This: Example 1C & 1D Multiply. Write the product as one power. C. x5 • y2 Cannot combine; the bases are not the same. x5 • y2 D. 412 • 417 41 2 + 7 Add exponents. 41 9

DIVIDING POWERS WITH THE SAME BASE Notice what occurs when you divide powers with the same base. 5 53 = 5  5  5 5  5  5  5  5 = 5 • 5 = 52 DIVIDING POWERS WITH THE SAME BASE Words Numbers Algebra To divide powers with the same base, keep the base and subtract the exponents. 6 5 9 – 4 9 4 = b m – n m n

Additional Example 2: Dividing Powers with the Same Base Divide. Write the product as one power. 7 5 3 A. 7 5 – 3 Subtract exponents. 7 2 x 10 9 B. x 10 – 9 Subtract exponents. x Think: x = x 1

A. B. 9 9 9 Subtract exponents. 97 e e e Subtract exponents. e Try This: Example 2 Divide. Write the product as one power. 9 9 A. 9 2 9 9 – 2 Subtract exponents. 97 e 10 B. e 5 e 10 – 5 Subtract exponents. e 5

This result can be confirmed by writing out the factors. When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0 exponent. 1 = 4 2 42 – 2 = 40 = 1 = This result can be confirmed by writing out the factors. = (4 • 4) = 1 1 4 2

0 does not exist because 00 represents a quotient of the form But the denominator of this quotient is 0, which is impossible, since you cannot divide by 0. Helpful Hint 0n .

THE ZERO POWER Words Numbers Algebra The zero power of any number except 0 equals 1. 1000 = 1 (–7)0 = 1 a0 = 1, if a  0

Additional Example 3: Astronomy Application A light-year, or the distance light travels in one year, is almost 1018 centimeters. To convert this number to kilometers, you must divide by 105. How many kilometers is a light-year? 10 18 5 10 18 - 5 Subtract exponents. 10 13 A light-year is almost 1013 km.

The ship had 104 metric tons of grain loaded. Try This: Example 3 A ship has 107 kilograms of grain loaded into its cargo hold. A metric ton is 103 kilograms. How many metric tons of grain were loaded? The weight in metric tons is equal to the weight in kilograms divided by 10 kilograms per metric ton. 3 10 7 3 10 7 - 3 Subtract exponents. 10 4 The ship had 104 metric tons of grain loaded.

Lesson Quiz: Part 1 Write the product or quotient as one power 1. n3  n4 n 7 2. 8 • 88 8 9 109 105 3. 10 4 t9 t7 t 2 4. 5. 33 • 32 • 35 3 10

Lesson Quiz: Part 2 6. A school would like to purchase new globes. They can get six dozen for $705.80 from Company A. From Company B, they can buy a half gross for $725.10. Which company should they buy from? (1 gross = 122 items) Company A