1. Exponents

2. What we want is to see the child in pursuit of knowledge, and not knowledge in pursuit of the child. George Bernard Shaw

3. EXPONENTS Exponents represents a mathematical shorthand that tells how many times a number is multiplied by itself.

4. Exponents Understanding exponents is important because this shorthand is used throughout subsequent mathematics courses. It appears often in formulas used in science, business, statistics, and geometry.

5. Location of Exponent An exponent is a little number high and to the right of a regular or base number. 3 4 Base Exponent

6. Definition of Exponent An exponent tells how many times a number is multiplied by itself. 3 4 Exponent

7. What an Exponent Represents An exponent tells how many times a number is multiplied by itself. 3 4 = 3 x 3 x 3 x 3 times 4

8. How to Read an Exponent This exponent is read: three to the fourth power 3 4 Base Exponent

9. Common Exponents This exponent is read: three to the second power or three squared 3 2 Exponent

10. Common Exponents This exponent is read: three to the third power or three cubed 3 3 Exponent

11. What is the Exponent? 2 x 2 x 2 =2 3

12. What is the Base and the Exponent? 8 x 8 x 8 x 8 =8 4

13. How to Multiply Out an Exponent (Standard Form) = 3 x 3 x 3 x 33 9 27 81 4

14. Write in Standard Form 4 2 = 16

15. Rules of Exponents Exponents come with their own set of rules Rules follow a natural emerging pattern

16. The factors of a power, such as 7 4, can be grouped in different ways. Notice the relationship of the exponents in each product. 7 × 7 × 7 × 7 = 7 4 (7 × 7 × 7) × 7 = 7 3 × 7 1 = 7 4 (7 × 7) × (7 × 7) = 7 2 × 7 2 = 7 4 Multiplication Rule

17. Multiply 3 4 × 3 3 3 4 × 3 3 = (3 × 3 × 3 × 3) × (3 × 3 × 3) = (3 × 3 × 3 × 3 × 3 × 3 × 3) = 3 7 Multiplication Rule 7 times

18. Words NumbersAlgebra To multiply powers with the same base, keep the base and add the exponents. b m × b n = b m + n 3 5 × 3 8 = 3 5 + 8 = 3 13 MULTIPLYING POWERS WITH THE SAME BASE Multiplication Rule

19. Examples Multiply and write the product as one power: 66 × 6366 × 63 6 9 6 6 + 3 4 5 × 4 7 4 12 4 5 + 7 Add exponents. Multiplication Rule

20. 24 4 × 24 4 2 5 × 2 2 6 2 5 + 1 24 8 4 + 4 Think: 2 = 2 1 Add exponents. Multiplication Rule Additional Examples Multiply and write the product as one power:

21. Notice what occurs when you divide powers with the same base. 5 5 5353 = 5 × 5 × 5 5 × 5 × 5 × 5 × 5 = 5 × 5 = 5 2 = 5 × 5 × 55 × 5 × 5 5 × 5 × 5 × 5 × 5 Division Rule

22. DIVIDING POWERS WITH THE SAME BASE WordsNumbersAlgebra To divide powers with the same base, keep the base and subtract the exponents. 6 5 6 9 – 4 6 9 6 4 = = b m – n b m b n = Division Rule

23. Subtract exponents. 7 2 7 5 – 3 7 5 7 3 Divide and w Write the product as one power 2 10 2 9 Subtract exponents. 2 10 – 9 2 Think: 2 = 2 1 Division Rule

24. When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0 exponent. This result can be confirmed by writing out the factors. 1 = 4 2 4 2 4 2 – 2 = 4 0 = 1 = = (4 × 4) = 1 1 1 = 4 2 2 = (4 × 4) 4 Zero Rule

25. THE ZERO POWER WordsNumbers Algebra The zero power of any number except 0 equals 1. 100 0 = 1 (–7) 0 = 1 a 0 = 1, if a  0 Zero Rule

26. ORDER OF OPERATIONS How to do a math problem with more than one operation in the correct order.

27. Order of Operations Problem:Evaluate the following arithmetic expression: 3 + 4 x 2 Solution: Student 1 3 + 4 x 2 = 7 x 2 = 14 Student 2 3 + 4 x 2 3 + 8 11

28. Order of Operations  It seems that each student interpreted the problem differently, resulting in two different answers.  Student 1 performed the operation of addition first, then multiplication  Student 2 performed multiplication first, then addition.

29. Order of Operations  When performing arithmetic operations there can be only one correct answer. We need a set of rules in order to avoid this kind of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation.

30. Order of Operations Rule 1:First perform any calculations inside parentheses. Rule 2:Next perform all multiplications and divisions, working from left to right. Rule 3:Lastly, perform all additions and subtractions, working from left to right.

31. Example Expression Evaluation Operation 6 + 7 x 8 = 6 + 7 x 8 Multiplication = 6 + 56 Addition = 62 16 ÷ 8 - 2 = 16 ÷ 8 - 2 Division = 2 - 2 Subtraction = 0 (25 - 11) x 3 = (25 - 11) x 3 Parentheses = 14 x 3 Multiplication = 42

32. Time to do some computing! Evaluate using the order of operations. 3 + 6 x (5 + 4) ÷ 3 - 7 Solution: Step 1: = 3 + 6 x 9 ÷ 3 - 7 Parentheses Step 2: = 3 + 54 ÷ 3 - 7 Multiplication Step 3: = 3 + 18 - 7 Division Step 4: = 21 - 7 Addition Step 5: = 14 Subtraction

33. 1) 5 + (12 – 3) 5 + 9 14 2) 8 – 3 2 + 7 8 - 6 + 7 2 + 7 9 3) 39 ÷ (9 + 4) 39 ÷ 13 3 Examples

34. Fractions Evaluate the arithmetic expression below: This problem includes a fraction bar, which means we must divide the numerator by the denominator. However, we must first perform all calculations above and below the fraction bar BEFORE dividing. The fraction bar can act as a grouping symbol

35. Thus Evaluating this expression, we get: Fractions

36. Mr. Smith charged Jill $32 for parts and $15 per hour for labor to repair her bicycle. If he spent 3 hours repairing her bike, how much does Jill owe him? Solution: 32 + 3 x 15= 32 + 3 x 15= 32 + 45 = 77 Jill owes Mr. Smith $77. Write an arithmetic expression

37. Add Parentheses to Obtain Result 7 − 1 + 6 = 0 7 − ( 1 + 6 ) = 0 1 + 2 × 5 + 6 = 21 ( 1 + 2 ) × 5 + 6 = 21 2 + 2 × 5 ÷ 3 − 1 = 10 ( 2 + 2 ) × 5 ÷ ( 3 − 1 ) = 10 3 + 8 ÷ 2 = 7 3 + ( 8 ÷ 2 ) = 7 8 + 3 − 7 = 4

38. When evaluating arithmetic expressions, the order of operations is: Simplify all operations inside parentheses. Perform all multiplications and divisions, working from left to right. Perform all additions and subtractions, working from left to right. If a problem includes a fraction bar, perform all calculations above and below the fraction bar before dividing the numerator by the denominator. Summary