Section 5.3 Negative 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 for and Use this rule to simplify: _____________

1. (b) Assume that the quotient rule, which states that Use this rule to simplify:_____________ foris 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 Algebraically For any nonzero real number x and natural number n, Verbally A nonzero base with a negative exponent can be rewritten by reciprocating the base and using the corresponding positive exponent. Algebraic Example

Simplify each expression. 2.3.

Simplify each expression. 4.5.

Note the effect of a negative exponent on a fraction. 6. Simplify ____________

Fraction to a Negative Power Algebraically Verbally Numerical Example For any nonzero real numbers x and y and natural number n, A nonzero fraction to a negative exponent can be rewritten by reciprocating the fraction and using the corresponding positive exponent.

Simplify each expression. 7.

Simplify each expression. 8.

Simplify each expression. 9.

Simplify each expression. 10.

Simplify each expression. 11.

Simplify each expression. 12.

Summary of the Exponent Rules: For any nonzero real numbers x and y and whole number exponents m and n, Product rule:_________ Power rule: Quotient rule: ____________ Zero exponent:____________for Negative exponent rule: _________

Simplify each expression to a form involving only positive exponents. Assume and 13.

Simplify each expression to a form involving only positive exponents. Assume and 14.

Simplify each expression to a form involving only positive exponents. Assume and 15.

Simplify each expression to a form involving only positive exponents. Assume and 16.

Simplify each expression to a form involving only positive exponents. Assume and 17.

Simplify each expression to a form involving only positive exponents. Assume and 18.

Evaluate each expression for x = 2 and y =

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

21. 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 a. The decimal point is moved 2 places to the right. b. The decimal point is not moved. c. The decimal point is moved 2 places to the left.

Write each number in standard decimal notation. 22.

Write each number in standard decimal notation. 23.

Write each number in standard decimal notation. 24.

Write each number in standard decimal notation. 25.

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

Write each number in scientific notation ,000

Write each number in scientific notation ,300

Write each number in scientific notation

Write each number in scientific notation

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

31. Each song on a personal music player requires about bytes of memory. If the music player has 80 GB ( bytes) of memory available, approximate the number of songs it will hold.

32. Use scientific notation to estimate (4,990,000)( ). Pencil and Paper Estimate: Calculator Approximation: