1. 4-1 Exponents Repeated Multiplication or Division Using Bases and Exponents

2. 4-1 Exponents Vocabulary/Essential Question EQ: How can we perform repeated multiplication and division in a shortened form? exponential form exponent base power

3. 4-1 Exponents If a number is in exponential form, the exponent represents how many times the base is to be used as a factor. A number produced by raising a base to an exponent is called a power. Both 27 and 3 3 represent the same power. 7 Exponent Base 2

4. 4-1 Exponents. Simple exponent examples

5. 4-3 Properties of Exponents 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

6. 4-1 Exponents Identify how many times 4 is a factor. 4 4 4 4 = 4 4 Write in exponential form. Additional Example 1: Writing Exponents A. 4 4 4 4 Read –(6 3 ) as “negative 6 to the 3rd power” or “negative 6 cubed”. Reading Math Identify how many times –6 is a factor. (–6) (–6) (–6) = (–6) 3 B. (–6) (–6) (–6) ‏

7. 4-1 Exponents Identify how many times 5 and d are used as a factor. Additional Example 1: Writing Exponents C. 5 5 d d d d Write in exponential form. 5 5 d d d d = 5 2 d 4

8. 4-1 Exponents. Bases with negative exponents or exponents with negative bases.

9. 4-1 Exponents Look for a pattern in the table to extend what you know about exponents to include negative exponents. ÷ 10 10 2 1 0 –1 10 –2 10 100 10 1 1 1 1 = 0.1 1 10 1 100 = 0.01

10. 4-1 Exponents D. = 256 = (–4) (–4) (–4) (–4) ‏ (–4) 4 C. (–4) 4 Simplify. Additional Example 2: Simplifying Powers Find the product of four –4’s. Find the product of eight 1/2’s.

11. 4-1 Exponents D. 9 7 = – 25 9 7 = 9 9 9 9 9 9 9 = 4,782,969 = –(5) (5) ‏ –(5) 2 C. –(5) 2 Simplify. Check It Out: Example 2 Find the product of two 5’s and then make the answer negative. Find the product of seven 9’s.

12. 4-1 Exponents Additional Example 1: Using a Pattern to Simplify Negative Exponents Simplify. Write in decimal form. A. 10 –2 10 –2 = 1 10 = 1 100 = 0.01 B. 10 –1 = = 0.1 1 10 = 1 Extend the pattern from the table. Multiply. Write as a decimal. Extend the pattern from the table. Multiply. Write as a decimal.

13. 4-1 Exponents Check It Out: Example 1A 10 –8 = 1 100,000,000 = 1 10 10 10 10 Extend the pattern from the table. = 0.00000001 Multiply. Write as a decimal. Simplify. Write in decimal form.

14. 4-1 Exponents 5 –3 Write the power under 1; change the sign of the exponent. Additional Example 2A: Evaluating Negative Exponents Simplify. Find the product of three ’s. 1515 Simplify.

15. 4-1 Exponents (–10) –3 Write the power under 1; change the sign of the exponent. Additional Example 2B: Evaluating Negative Exponents Simplify. Find the product of three ’s. 1 –10 Simplify. 1 –10 –10 –10 –1000 1 = –0.001

16. 4-1 Exponents. Multiplying powers with the same base.

17. 4-3 Properties of Exponents Additional Example 1: Multiplying Powers with the Same Base A. 6 6 6 3 6 9 6 6 + 3 B. n 5 n 7 n 12 n 5 + 7 Add exponents. Multiply. Write the product as one power.

18. 4-3 Properties of Exponents D. 24 4 24 4 C. 2 5 2 2 6 2 5 + 1 24 8 4 + 4 Think: 2 = 2 1 Additional Example 1: Multiplying Powers with the Same Base Continued Multiply. Write the product as one power. Add exponents.

19. 4-3 Properties of Exponents Check It Out: Example 1 A. 4 2 4 4 4 6 4 2 + 4 B. x 2 x 3 x 5 x 2 + 3 Add exponents. Multiply. Write the product as one power.

20. 4-3 Properties of Exponents Notice what occurs when you divide powers with the same base. 5  5  55  5  5 5 5 5353 = 5  5  5  5  5 = 5  5 = 5 2 = 5  5  55  5  5 5  5  5  5  5

21. 4-3 Properties of Exponents. Dividing powers with the same base.

22. 4-3 Properties of Exponents Subtract exponents. 7 2 7 5 – 3 7 5 7 3 Additional Example 2: Dividing Powers with the Same Base Divide. Write the quotient as one power. A. x 10 x 9 B. Subtract exponents. x 10 – 9 x Think: x = x 1

23. 4-3 Properties of Exponents Subtract exponents. 9797 9 9 – 2 9 9 9 2 Check It Out: Example 2 Divide. Write the product as one power. A. B. e 10 e 5 Subtract exponents. e 10 – 5 e 5

24. 4-3 Properties of Exponents. Raising a power to a power

25. 4-3 Properties of Exponents Simplify. Multiply exponents. Additional Example 3: Raising a Power to a Power A. (5 4 ) 2 (5 4 ) 2 5 4 2 5858 B. (6 7 ) 9 (6 7 ) 9 6 7 9 6 63 Multiply exponents.

26. 4-3 Properties of Exponents Simplify. Multiply exponents. Additional Example 3: Raising a Power to a Power C. D. (17 2 ) –20 (17 2 ) –20 17 2 –20 17 –40 Multiply exponents. 2323 12–3 2323 12 –3

27. 4-3 Properties of Exponents Simplify. Multiply exponents. Check It Out: Example 3 A. (3 3 ) 4 (3 3 ) 4 3 3 4 3 12 B. (4 8 ) 2 (4 8 ) 2 4 8 2 4 16 Multiply exponents.

28. 4-3 Properties of Exponents Simplify. Multiply exponents. Check It Out: Example 3 C. D. (13 4 ) –10 (13 4 ) –10 13 4 –10 13 –40 Multiply exponents. 1414 11–2 1414 11–2

29. 4-3 Properties of Exponents Lesson Quiz Write the product or quotient as one power. 3. 8 9 n 7 1. n 3  n 4 10 9 10 5 10 4 4. t 2 5. 3 2 3 3 3 5 3 10 2. 8 8 8 t 9 t 7 6. (m 2 ) 19 m 38 7. (9 -8 ) 9 9 –72 8. (10 4 ) 0 1

30. 4-3 Properties of Exponents 2. Write the product as one power. 4 4  4 3  4 2 A. 4 9 B. 4 14 C. 4 24 D. 4 432 Lesson Quiz for Student Response Systems

31. 4-1 Exponents Additional Example 3: Using the Order of Operations 4(7) + 16 Substitute 4 for x, 2 for y, and 3 for z. Evaluate the exponent. Subtract inside the parentheses. Multiply from left to right. 4(2 4 – 3 2 ) + 4 2 4(16 – 9) + 16 28 + 16 Evaluate x(y x – z y ) + x for x = 4, y = 2, and z = 3. y x(y x – z y ) + x y Add. 44

32. 4-1 Exponents Check It Out: Example 3 60 – 7(7) Substitute 5 for x, 2 for y, and 60 for z. Evaluate the exponent. Subtract inside the parentheses. Multiply from left to right. 60 – 7(2 5 – 5 2 ) ‏ 60 – 7(32 – 25) 60 – 49 Evaluate z – 7(2 x – x y ) for x = 5, y = 2, and z = 60. z – 7(2 x – x y ) Subtract. 11

33. 4-1 Exponents Subtract inside the parentheses. Evaluate 5 – (6 – 4) –3 + (– 2) 0. Evaluate the exponents. Additional Example 3: Using the Order of Operations 5 – (6 – 4) –3 + (–2) 0 = 5 – (2) –3 + (–2) 0 = 5 – + 1 1818 = 5 7878 Add and subtract from left to right.

34. 4-1 Exponents Subtract inside the parentheses. Evaluate 3 + (7 – 4) –2 + (–8) 0. Evaluate the exponents. Check It Out: Example 3 3 + (7 – 4) –2 + (–8) 0 = 3 + (3) –2 + (–8) 0 = 3 + + 1 1919 = 4 1919 Add.

35. 4-3 Properties of Exponents Problem of the Day Calculate 6 to the fourth power minus 56. 1240