Exponents and Scientific Notation P.2

1. Exponents and Scientific NotationP.2 EQ: Describe the properties of exponents

2. Definition of a Natural Number Exponent • If b is a real number and n is a natural number, • bn is read “the nth power of b” or “ b to the nth power.” Thus, the nth power of b is defined as the product of n factors of b. Furthermore, b1 = b

3. The Negative Exponent Rule • If b is any real number other than 0 and n is a natural number, then

4. The Zero Exponent Rule • If b is any real number other than 0, b0 = 1.

5. b m · b n = b m+n When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base. The Product Rule

6. (bm)n = bm•n When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses. The Power Rule (Powers to Powers)

7. The Quotient Rule • When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.

8. Example • Find the quotient of 43/42 Solution:

9. (ab)n = anbn When a product is raised to a power, raise each factor to the power. Products to Powers

10. Text Example • -16y4 • -8y4 • 16y4 • 8y4 Simplify: (-2y)4. Solution (-2y)4 = (-2)4y4 = 16y4

11. Quotients to Powers • When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.

12. Example • Simplify by raising the quotient (2/3)4 to the given power. Solution:

13. Scientific Notation The number 5.5 x 1012 is written in a form called scientific notation. A number in scientific notation is expressed as a number greater than or equal to 1 and less than 10 multiplied by some power of 10. It is customary to use the multiplication symbol, x, rather than a dot in scientific notation.

14. Text Example • Write the following number in decimal notation: 2.6 X 107 Solution: a. 2.6 x 107 can be expressed in decimal notation by moving the decimal point in 2.6 seven places to the right. We need to add six zeros. 2.6 x 107 = 26,000,000.

15. Write the following number in decimal notation 1.016 X 10-8 b. 1.016 x 10-8 can be expressed in decimal notation by moving the decimal point in 1.016 eight places to the left. We need to add seven zeros to the right of the decimal point. 1.016 x 10-8 = 0.00000001016.

16. Decimal point moves 6 places a. 4,600,000 = 4.6 x 10? 4.6 x 106 Decimal point moves 4 places b. 0.00023 = 2.3 x 10? 2.3 x 10-4 Text Example Write each number in scientific notation. a. 4,600,000 b. 0.00023 Solution