Section 5.3Negative Exponents and Scientific Notation

5.3 Lecture Guide: Negative Exponents and Scientific Notation

Objective: Simplify expressions with negative exponents.

1.(a) In the previous section, we stated the quotient rule as

1m

n n m

x

x x for 0x and .n m

Use this rule to simplify: 4

7

x

x_____________

1.(b) Assume that the quotient rule, which states that

Use this rule to simplify: 4

7

x

x_____________

mm n

n

xx

x for 0,x is true for all integral values

of m and n.

1.(c) Since we want the two expressions above to be equal we have ___________ = __________.

Generalizing, we get the following result for negative exponents:

Negative Exponents

AlgebraicallyFor any nonzero real number x and natural number n,

1nn

xx

VerballyA nonzero base with a negative exponent can be rewritten by reciprocating the base and using the corresponding positive exponent.

Algebraic Example4x

Simplify each expression.

2. 42 3. 42

Simplify each expression.

4. 5.1 13 5 1(3 5)

Note the effect of a negative exponent on a fraction.

6. Simplify 1

2 1233

____________

Fraction to a Negative Power

Algebraically

Verbally

Numerical Example

For any nonzero real numbers x and y and natural

number n,

n nx y

y x

A nonzero fraction to a negative exponent can be rewritten by reciprocating the fraction and using the corresponding positive exponent.

22

7

Simplify each expression.1

10

3

7.

Simplify each expression.

8.3

25

Simplify each expression.

9.223

Simplify each expression.

10.2

23

Simplify each expression.

11.2xy

Simplify each expression.

12.2

xy

Summary of the Exponent Rules:

For any nonzero real numbers x and y and whole number exponents m and n,

Product rule: m nx x _________

Power rule:

( )mxy

Quotient rule: m

n

x

x ____________

Zero exponent: 0x ____________ for 0x

Negative exponent rule: 1nnxx

_________( )m nx m

x

y

_________ _________

Simplify each expression to a form involving only positive exponents.

Assume 0x and 0.y

13.

45

3

xx

Simplify each expression to a form involving only positive exponents.

Assume 0x and 0.y

14. 22 43 5x x

Simplify each expression to a form involving only positive exponents.

Assume 0x and 0.y

15.2 3

1 4

x yx y

Simplify each expression to a form involving only positive exponents.

Assume 0x and 0.y

16.

23 5

43

5

3

x y

xy

Simplify each expression to a form involving only positive exponents.

Assume 0x and 0.y

17.3 7

2 4

36 1512 45x xx x

Simplify each expression to a form involving only positive exponents.

Assume 0x and 0.y

18.

23 2

2 4

1435x yx y

Evaluate each expression for x = 2 and y = 3.

19.2 2x y

20. 2x y

Evaluate each expression for x = 2 and y = 3.

21. 2( )x y

Evaluate each expression for x = 2 and y = 3.

Objective: Use scientific notation.

Verbally

Writing a Number in Standard Decimal Notation

Multiply out the two factors by using the given power of ten. a. If the exponent on 10 is positive, move the decimal point to the right.

b. If the exponent on 10 is zero, do not move the decimal point.

c. If the exponent on 10 is negative, move the decimal point to the left.

Numerical Examples

23.456 10 3.456 100 345.6 a.The decimal point is moved 2 places to the right.

b.The decimal point is not moved.

03.456 10 3.456 1 3.456

c.The decimal point is moved 2 places to the left.

23.456 10 3.456 0.01 0.03456

Write each number in standard decimal notation.

22. 45.71 10

Write each number in standard decimal notation.

23. 44.25 10

Write each number in standard decimal notation.

24. 63.2 10

Write each number in standard decimal notation.

25. 73.987 10

Writing a Number in Scientific Notation:

Verbally 1. Move the decimal point immediately to the right of the first nonzero digit of the number.

2. Multiply by a power of 10 determined by counting the number of places the decimal point has been moved.

a. The exponent on 10 is 0 or positive if the magnitude of the original number is 1 or greater.

Numerical Examples: 03.456 3.456 10 2345.6 3.456 10

b. The exponent on 10 is negative if the magnitude of the original number is less than 1.

Numerical Examples: 20.03456 3.456 10

Write each number in scientific notation.

26. 80,000

Write each number in scientific notation.

27. 72,300

Write each number in scientific notation.

28. 0.008

Write each number in scientific notation.

29. 0.0000985

30. Write the result on the calculator screen in scientific notation and in standard decimal notation. See Calculator Perspective 5.3.1.

Scientific notation:

Standard decimal notation:

31. Each song on a personal music player requires about 64 10 bytes of memory. If the music player has 80 GB

( 108.0 10 bytes) of memory available, approximate thenumber of songs it will hold.

32. Use scientific notation to estimate (4,990,000)(0.000147).

Pencil and Paper Estimate:

Calculator Approximation: