The document discusses exponents and the rules of exponentiation. It defines exponents, bases, and examples of exponents. It then outlines five rules of exponentiation and uses examples to illustrate each rule. It concludes by defining exponents of 0 and negative exponents.

1. 1.3
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Exponents
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2. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Goal
To learn about exponents and
the rules of exponentiation.
MATH1003

3. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Exponents
n is called the
b is called the Definition:
exponent
Definition
base
bn = b • b • b • b • … • b
where b occurs n times
Example:
48 = 4 • 4 • 4 • 4 • 4 • 4 • 4 • 4
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4. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
Is 42 • 43 equal to 4(2 + 3)? Yes
42 • 43 4(2 + 3)
= 16 • 64 = 45
= 1024 = 1024
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5. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
So, 42 • 43 = 4(2 + 3)
Rule 1 Rule 1:
am • an = am + n
MATH1003

6. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
35 equal to 3(5 - 2)?
Is 2 Yes
3
3 5
3(5 - 2)
32 = 33
243
= 9 = 27
= 27
MATH1003

7. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
35
So, 32 = 3(5 - 2)
Rule 2 Rule 2:
am = am - n
a n
MATH1003

8. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
Is (24)2 equal to 2(4 • 2)? Yes
(24)2 2(4 • 2)
= 24 • 2 4 = 28
= 28 from Rule 1 = 256
= 256
MATH1003

9. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
So, (24)2 = 2(4 • 2)
Rule 3 Rule 3:
(am)n = amn
MATH1003

10. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
Is (3y)2 equal to 32y2? Yes
(3y)2
= (3y) • (3y) we know that
multiplication is
= 3 • y • 3 • y commutative
= 3 • 3 • y • y
= 32y2
MATH1003

11. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
So, (3y)2 = 32y2
Rule 4 Rule 4:
(ab)n = anbn
MATH1003

12. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
2
Is()
x equal to x2 ?
3 3 2 Yes
2
(3) = 3 • 3
x x x
x•x
= 3•3
x2
= 32
MATH1003

13. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
2
So, x
()
x2
= 2
3 3
Rule 5
()
Rule 5:
a
b
n
an
= bn
MATH1003

14. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Exponents
Definition
for all real
numbers as long as Definition:
a≠0
a0 = 1
Examples:
30 = 1
-65.530 = 1
10000302329178273130 = 1
MATH1003

15. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Exponents
Definition
for all real
Definition: numbers as long as
-n = 1
a≠0
a an
Examples:
2 -4 = 1
2 4
32 (2-5) = 3-3 = 1
3 5 = 3 33
MATH1003