Recall that multiplication with the same base results in addition of exponents; that is, a r a s = a r+s.

Transcription

1 Section 5.2

2 Subtract Simplify

3 Recall that multiplication with the same base results in addition of exponents; that is, a r a s = a r+s. Since division is the inverse operation of multiplication, we can expect division with the same base to result in subtraction of exponents.

4 Negative Exponents

5 To develop the properties for exponents under division, we again apply the definition of exponents: Notice: 5 3 = 2 Notice: 7 4 = 3

6 In both cases division with the same base resulted in subtraction of the smaller exponent from the larger.

7 Write each expression with a positive exponent and then simplify: Notice: Negative exponents do not indicate negative numbers. They indicate reciprocals

8 Property 4 and the Properties of Exponents

9 We know that Let's decide now that with division of the same base, we will always subtract the exponent in the denominator from the exponent in the numerator. Subtracting the bottom exponent from the top exponent Subtraction Definition of negative exponents We get the same result which leads to Property 4.

10 We can now continue the list of properties of exponents.

11 Simplify the following expressions: x x 6 = x x x = x The position of the term with the larger exponent gives us the position of the exponential term in the quotient =

12 Property 5 and the Properties of Exponents

13 Our final property of exponents is similar to Property 3, but it involves division instead of multiplication.

14 Simplify the following expressions. x 2 3 = x y 2 = 2 5 y =

15 Zero and One as Exponents

16 We have two special exponents : 0 and 1. To demonstrate, we will solve a problem two different ways: Hence x 1 = x Raising a number to the 1 st power preserves the number s identity. Stated generally, a 1 = a

17 We use the same procedure to obtain an expression for x 0 : Hence 5 0 = 1 Therefore, x 0 is equal to 1 with one exception. In the case of x = 0, we have 0 0, which we will not define. Stated generally, a 0 = 1 for all real numbers except a = 0.

18 Simplify the following expressions: 8 0 = = = = 5 (2x 2 y) 0 = 1

19 Combinations of Properties

20 Here is a summary of the definitions and properties of exponents we have developed so far. For each definition or property in the list, a and b are real numbers, and r and s are integers.

21 Simplify the expression. Write the answer with a positive exponent: = 25x 4 x 6 Property 2 A Power Raised to a Power Property 3 The Power of a Product Property 4 Division with the Same Base

22 We can use division to compare the area of geometric figures. In the following diagram, the length of a side of the larger square is 3 times as long as the length of a side of the smaller square. How many of the smaller squares will it take to cover up the larger square?

23 If we let x represent the length of a side of the smaller square, then the length of a side of the larger square is 3x. Square 1: A = x 2 Square 2: A = (3x) 2 = 9x 2

24 To find out how many smaller squares it will take to cover up the larger square, we divide the area of the larger square by the area of the smaller square. It will take 9 of the smaller squares to cover the larger square.

25 More on Scientific Notation

26 Now that we have completed our list of definitions and properties of exponents, we can expand the work we did previously with scientific notation. Recall that a number is in scientific notation when it is written in the form n 10 r where 1 n < 10 and r is an integer.

27 Since negative exponents give us reciprocals, we can use negative exponents to write very small numbers in scientific notation. For example, the number , when written in scientific notation, is equivalent to Here s why:

28 The table below lists some other numbers in both scientific notation and expanded form.

29 Notice that when the number is written in scientific notation, the decimal point in the first number is placed so that the number is between 1 and 10. The exponent on 10 in the second number keeps track of the number of places we moved the decimal point in the original number to get a number between 1 and 10: 376,000 =

30 = Moved 3 places. Keeps track of the 3 places we moved the decimal point.

31 Section 5.2 Page # 1, 3, 7, 9, 11, 13, 15, 19, 21, 25, 29, 35, 37, 41, 45, 59, 53, 57, 61, 71, 73, 75, 79, 81, 83