2. Lessons from the Math Zone: Exponents Click to Start Lesson

3. Lessons from the Math Zone Exponents © Copyright 2006, Don Link Permission Granted for Educational Use Only

4. 12345 33333++++ 35× 33333++++ ? 36× 3+ = 15 = 18 Repeated Addition Arithmetic Shortcuts

5. 12345 33333++++ 35× 33333++++ 36× 3+ 12345 33333×××× 35^ 33333×××× ? 36^ 3× = 15 = 18 = 243 = 729 Repeated Addition Repeated Multiplication “caret” Arithmetic Shortcuts ANALOGY

6. 33333×××× 39^ 3× = 19,683 33××3× Another Example ^ 3^9 19683.

7. 33333×××× 39^ 3× = 19,683 33××3× 22222×××× 29^ 2× = 512 22××2× Another Example

8. BaseExponent^ 39^ The E EE Exponent tells us how many copies of the B BB Base to m mm multiply together. 39^ Multiply 9 99 9 copies of 3 33 3 together. 3 Base = 3 9 Exponent = 9 Terminology (or Power)

9. Let’s Practice with Calculators 3^63^6 = 729 5^65^6 = 15,625 7^77^7 = 823,543 10 ^4 = 10,000 15 ^7 = 170,859,375 2 ^20 = 1,048,576 20 ^5 = 3,200,000 1.5 ^4 = 5.0625 0.2 ^4 = 0.0016 10.2 ^4 = 10,824.3216

10. Let’s Practice with Calculators 3^63^6 = 729 5^65^6 = 15,625 7^77^7 = 823,543 10 ^4 = 10,000 15 ^7 = 170,859,375 2 ^20 = 1,048,576 20 ^5 = 3,200,000 1.5 ^4 = 5.0625 0.2 ^4 = 0.0016 10.2 ^4 = 10,824.3216 Good Job!

11. Let’s Practice without Calculators 3^23^2 = 9 5^35^3 = 125 7^37^3 = 343 10 ^3 = 1,000 15 ^1 = 15 2^62^6 = 64 1^91^9 = 1 0^40^4 = 0 3^43^4 = 81 4^34^3 = 64

12. Let’s Practice without Calculators 3^23^2 = 9 5^35^3 = 125 7^37^3 = 343 10 ^3 = 1,000 15 ^1 = 15 2^62^6 = 64 1^91^9 = 1 0^40^4 = 0 3^43^4 = 81 4^34^3 = 64 Excellent!

13. Finding the Correct Exponent 5×5×5×5×5×55×5×5×5×5×5 = 5^__ 6 12345 6 8×8×8×88×8×8×8 = 8^__ 4 2×2×2×2×2×2×22×2×2×2×2×2×2 = 2^__ 7 7×77×7 = 7^__ 2 1.5 × 1.5 × 1.5 × 1.5 × 1.5 = 1.5^__ 5 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 4^__ 11 Base Exponent

14. Finding the Correct Exponent 5×5×5×5×5×55×5×5×5×5×5 = 5^__ 6 123456 8×8×8×88×8×8×8 = 8^__ 4 2×2×2×2×2×2×22×2×2×2×2×2×2 = 2^__ 7 7×77×7 = 7^__ 2 1.5 × 1.5 × 1.5 × 1.5 × 1.5 = 1.5^__ 5 4×4×4×4×4×4×4×4×4×4×4×44×4×4×4×4×4×4×4×4×4×4×4 = 4^__ 13 Base Exponent Cookin’!

15. 9393^ “Three to the n nn ninth power” 5 4 “Five to the f ff fourth power” 7 2 “Seven to the s ss second power” “or S SS Seven s ss squared” 10 3 “Ten to the t tt third power” “or T TT Ten c cc cubed” Exponents without the Caret

16. 5 4 Let’s Practice with Calculators = 625 9 4 = 6,561 4 9 = 262,144 17 6 = 24,137,569 3 12 = 531,441 7 7 = 823,543 0.5 4 = 0.0625 2.5 4 = 39.0625 122.5 2 = 15,006.25

17. 5 4 Let’s Practice with Calculators = 625 9 4 = 6,561 4 9 = 262,144 17 6 = 24,137,569 3 12 = 531,441 7 7 = 823,543 0.5 4 = 0.0625 2.5 4 = 39.0625 122.5 2 = 15,006.25 Yes Indeed!

18. 5 2 Let’s Practice without Calculators = 25 9 3 = 729 4 4 = 256 17 2 = 289 3 1 = 3 7 4 = 2,401 0 4 = 0 1 14 = 1 0 x =0

19. 5 2 Let’s Practice without Calculators = 25 9 3 = 729 4 4 = 256 17 2 = 289 3 1 = 3 7 4 = 2,401 0 4 = 0 1 14 = 1 2 7 = 128 Right On!

20. Finding the Correct Exponent 5×5×5×5×55×5×5×5×5 = 5 5 12345 8×8×8×88×8×8×8 = 8 4 2×2×2×2×2×22×2×2×2×2×2 = 2 6 6×66×6 = 6 2 1.5 × 1.5 × 1.5 × 1.5 × 1.5 = 1.5 5 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 4 12 Base Exponent ?

21. Finding the Correct Exponent 5×5×5×5×55×5×5×5×5 = 5 5 12345 8×8×8×88×8×8×8 = 8 4 2×2×2×2×2×22×2×2×2×2×2 = 2 6 6×66×6 = 6 2 1.5 × 1.5 × 1.5 × 1.5 × 1.5 = 1.5 5 4×4×4×4×4×4×4×4×4×4×44×4×4×4×4×4×4×4×4×4×4 = 4 12 Base Exponent ? High Five!

22. 1 What if the exponent is zero? = 81 = 27 = 9 = 3 = Let’s Follow a Pattern = –1–1 –1–1 ÷3÷3 ÷3÷3 –1–1 ÷3÷3 –1–1 ÷3÷3 1 = 1 CLICK for EXTENSION: Negative Exponents CLICK for EXTENSION: Negative Exponents CLICK for EXTENSION: 0 CLICK for EXTENSION: 0 ?

23. Name? Base Exponents: Summary and Review Name? Exponent (or Power) On calculator Name? Caret = 3× 3×3×3 fourth power “Three to the fourth power” cubed “Three cubed” squared “Three squared”

24. For Printing

25. Extension: Negative Exponents = 9 = 3 = 1 Let’s Extend the Pattern –1–1 ÷3÷3 –1–1 ÷3÷3 –1–1 ÷3÷3 = 1/3 –1–1 ÷3÷3 = 1/9

26. (-) 5 –2–2 Let’s Practice = 0.04 (1/25) With Calculators Use the (-) Key 5^ ־2 0.04

27. = 0.125 (1/8) 5 –2–2 Let’s Practice = 0.04 (1/25) 2 –3–3 10 –5–5 = 0.00001 0.05 –6–6 = 64,000,000 3 –3–3 = 0.037 With Calculators 037037… “A repeating decimal”

28. = 0.125 (1/8) 5 –2–2 Let’s Practice = 0.04 (1/25) 2 –3–3 10 –5–5 = 0.00001 0.05 –6–6 = 64,000,000 3 –3–3 = 0.037 1 –4–4 = 1 1 2 –1–1 = 1/2 With Calculators No Calculators 037037… 10 –5–5 = 0.00001 4 –2–2 = 1/16 10 –3–3 = 1/1000 3 –3–3 = 1/27 –12

29. 5 –2–2 Let’s Practice = 0.04 (1/25) 2 –3–3 = 0.125 (1/8) 10 –5–5 = 0.00001 0.05 –6–6 = 64,000,000 3 –3–3 = 0.037 1 –4–4 = 1 1 –14 = 1 2 –1–1 = 1/2 With Calculators No Calculators 037037… 10 –5–5 = 0.00001 4 –2–2 = 1/16 10 –3–3 = 1/1000 3 –3–3 = 1/27 Fantastic! Click to RETURN to Main Lesson Click to RETURN to Main Lesson

30. Extension: Zero to the Zero Power? 0 0 ? What does this mean? Rule 1: x 0 = 1 Rule 2: = 0 0 x Anything to the 0 power = 1. Zero to any power = 0. Which rule should we use?

31. Extension: Zero to the Zero Power? 0 0 ? What does this mean? When mathematicians have two perfectly good rules that give different answers for some problem like 00, they say the answer is __________ for this case. undefined So, is undefined! 00 Click to RETURN to Main Lesson Click to RETURN to Main Lesson