1. Exponents and Scientific Notation MATH 017 Intermediate Algebra S. Rook

2. 2 Overview Section 5.1 in the textbook –Product rule for exponents –Expressions raised to the 0 power –Quotient rule for exponents –Expressions raised to negative powers –Scientific notation

3. Product Rule

4. 4 Consider x 4 ∙ x 5 x∙x∙x∙x ∙ x∙x∙x∙x∙xx∙x∙x∙x ∙ x∙x∙x∙x∙x x9x9 Product Rule: x a ∙ x b = x a+b –When multiplying LIKE BASES, add the exponents –Only applies when the operation is multiplication

5. 5 Product Rule (Example) Ex 1: Simplify: (4xy 2 )(2x 2 y 3 )

6. 6 Product Rule (Example) Ex 2: Simplify: (-x 2 y 5 z)(7x 4 z 3 )

7. Expressions Raised to the 0 Power

8. 8 Consider x 0 –As long as x ≠ 0, x 0 = 1 –x can also be an expression

9. 9 Expressions Raised to the 0 Power (Example) Ex 3: (2w) 0

10. 10 Expressions Raised to the 0 Power (Example) Ex 4: -(x 2 y 3 z 2 ) 0

11. Quotient Rule

12. 12 Quotient Rule Consider x 5 / x 2 x∙x∙x∙x∙x / x∙xx∙x∙x∙x∙x / x∙x x3x3 Quotient Rule: x a / x b = x a-b –When dividing LIKE BASES, subtract the exponents –Only applies when the operation is division

13. 13 Quotient Rule (Example) Ex 5: Simplify:

14. 14 Quotient Rule (Example) Ex 6: Simplify:

15. Expressions with Negative Exponents

16. 16 Expressions with Negative Exponents Consider x 2 / x 6 x -4 by the quotient rule x∙x / x∙x∙x∙x∙x∙x 1 / x 4 We NEVER leave an expression with a negative exponent Flipping an exponent and its base from the numerator into the denominator (or vice versa) reverses the sign of the exponent

17. 17 Expressions with Negative Exponents (Continued) x -4 = x -4 / 1 = 1 / x 4 2 -3 = 2 -3 / 1 = 1 / 2 3 = 1 / 8 ≠ -8 –The sign of the exponent DOES NOT affect the sign of the coefficient (or base) –Whenever using the quotient rule, the initial result goes into the numerator

18. 18 Expressions with Negative Exponents (Example) Ex 7: Simplify – leave NO negative exponents:

19. 19 Expressions with Negative Exponents (Example) Ex 8: Simplify – leave NO negative exponents:

20. Scientific Notation

21. 21 Scientific Notation Scientific Notation: any number in the form of a x 10 b where -10 < a < 10, a ≠ 0 and b is an integer –One non-zero number to the left of the decimal point – the rest to the right –Count how many places and in which direction the decimal is moved If to the left, b is positive If to the right, b is negative

22. 22 Scientific Notation (Example) Ex 9: Write in scientific notation: 0.000135

23. 23 Scientific Notation (Example) Ex 10: Write in scientific notation: 451,000

24. 24 Standard Notation Standard Notation: writing a number with a product of a power of ten without the power of ten –Take the decimal and move it: To the right if b is positive To the left if b is negative Fill in empty spots with zeros

25. 25 Standard Notation (Example) Ex 11: Write in standard notation: 1.155 x 10 4

26. 26 Standard Notation (Example) Ex 12: Write in standard notation: 29.3 x 10 -3

27. 27 Summary After studying these slides, you should know how to do the following: –Apply the product rule when multiplying like bases –Evaluate expressions raised to the 0 power –Apply the quotient rule when dividing like bases –Simplify expressions raised to negative powers –Convert back and forth between scientific and standard notation