1. Lessons from
the
Math Zone
Exponents

2. Arithmetic Shortcuts
Repeated Addition 3 + 3 + 3 + 3 + 3 1 2 3 4 5 3 × ? 5
= 15 3 × 6
= 18 3 + 3 + 3 + 3 + 3 + 3

3. Arithmetic Shortcuts
Repeated Addition
Repeated Multiplication 3 + 3 + 3 + 3 + 3 3 × 3 × 3 × 3 × 3
ANALOGY 1 2 3 4 5 1 2 3 4 5 3 × 5
= 15 3 ^ 5 ?
= 243
“caret” 3 × 6
= 18 3 ^ 6
= 729 3 + 3 + 3 + 3 + 3 + 3 3 × 3 × 3 × 3 × 3 × 3

4. 3 ^ 9 = 19,683 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 ^ Another Example 3^9
19683. 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 ^

5. Another Example 3 ^ 9
= 19,683 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 2 ^ 9
= 512 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

6. 3 ^ 9 Base ^ Exponent (or Power) 3 ^ 9 Terminology
The Exponent tells us how many copies of the Base to multiply together.
Base = 3 3 ^ 9
Multiply 9 copies of 3 together.
Exponent = 9

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

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

9. Let’s Practice without Calculators
3^2
= 9
0^4
= 0
5^3
= 125
3^4
= 81
7^3
= 343
4^3
= 64
10^3
= 1,000
15^1
= 15
2^6
= 64
1^9
= 1

10. Let’s Practice without Calculators
3^2
= 9
0^4
= 0
5^3
= 125
3^4
= 81
7^3
= 343
4^3
= 64
10^3
= 1,000
15^1
= 15
2^6
= 64
1^9
= 1

11. Finding the Correct Exponent1 2 3 4 5 6
5×5×5×5×5×5
= 5^__ 6
Base
8×8×8×8
= 8^__ 4
2×2×2×2×2×2×2
= 2^__ 7
7×7
= 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

12. Finding the Correct Exponent1 2 3 4 5 6
5×5×5×5×5×5
= 5^__ 6
Base
8×8×8×8
= 8^__ 4
2×2×2×2×2×2×2
= 2^__ 7
7×7
= 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
= 4^__ 13

13. Exponents without the Caret3 3 ^ 9 9
“Three to the ninth power” 4 5
“Five to the fourth power” 2 7
“Seven to the second power”
“or Seven squared” 3 10
“Ten to the third power”
“or Ten cubed”

14. Let’s Practice with Calculators4 4 5
= 625
0.5 = 4 4 9
= 6,561
2.5 = 9 2 4
= 262,144
122.5
= 15,006.25 6 17
= 24,137,569 12 3
= 531,441 7 7
= 823,543

15. Let’s Practice with Calculators4 4 5
= 625
0.5 = 4 4 9
= 6,561
2.5 = 9 2 4
= 262,144
122.5
= 15,006.25 6 17
= 24,137,569 12 3
= 531,441 7 7
= 823,543

16. Let’s Practice without Calculators2 4 5
= 25
= 0 3 x = 9
= 729 4 4
= 256 14 1
= 1 2 17
= 289 1 3
= 3 4 7
= 2,401

17. Let’s Practice without Calculators2 4 5
= 25
= 0 3 14 9
= 729 1
= 1 4 7 4
= 256 2
= 128 2 17
= 289 1 3
= 3 4 7
= 2,401

18. Finding the Correct Exponent1 2 3 4 5
5×5×5×5×5
= 5 5 ?
Base
8×8×8×8
= 8 4 6
2×2×2×2×2×2
= 2 2
6×6
= 6 5
1.5×1.5×1.5×1.5×1.5
= 1.5 12
4×4×4×4× 4×4×4×4× 4×4×4×4
= 4

19. Finding the Correct Exponent1 2 3 4 5
5×5×5×5×5
= 5 5 ?
Base
8×8×8×8
= 8 4 6
2×2×2×2×2×2
= 2 2
6×6
= 6 5
1.5×1.5×1.5×1.5×1.5
= 1.5 12
4×4×4×4×4×4×4×4×4×4×4
= 4

20. What if the exponent is zero? CLICK for EXTENSION: Negative Exponents
Let’s Follow a Pattern
= 81 = ? –1 ÷3
= 27 –1 ÷3
= 1
= 9 –1 ÷3
= 3 –1 ÷3 = 1 1
CLICK for EXTENSION: Negative Exponents
CLICK for EXTENSION: 00

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

22. For Printing

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

24. 5 = 0.04 (1/25) Let’s Practice –2 With Calculators 5^ ־2 0.04 Use the
(-) Key
5^ ־2
0.04
(-)

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

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

27. Let’s Practice
With Calculators
No Calculators –2 –4 5
= 0.04 (1/25) 1
= 1 –3
–14 2
= (1/8) 1
= 1 –5 –1 10 = 2
= 1/2 –6 –2
0.05
= 64,000,000 4
= 1/16 –3 3 –3
= 0.037
037037… 3
= 1/27 –5 10 –3 = 10
= 1/1000

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

29. Extension: Zero to the Zero Power? Click to RETURN to Main Lesson?
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.
So, is undefined!
undefined
Click to RETURN to Main Lesson