math1003 1.3 - exponents
TRANSCRIPT
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
1.3Exponents
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Goal
To learn about exponents andthe rules of exponentiation.
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Exponents
Definition:
bn = b • b • b • b • … • bwhere b occurs n times
Example:
48 = 4 • 4 • 4 • 4 • 4 • 4 • 4 • 4
b is called the base
n is called the exponent
Definition
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
Is 42 • 43 equal to 4(2 + 3)?
42 • 43
= 16 • 64= 1024
4(2 + 3)
= 45
= 1024
Yes
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
So, 42 • 43 = 4(2 + 3)
Rule 1:
am • an = am + nRule 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
== 27
3(5 - 2)
= 33
= 27
YesIs equal to 3(5 - 2)?35
32
35
32
2439
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
So, = 3(5 - 2)
Rule 2:
= am - nam
an
35
32
Rule 2
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
Is (24)2 equal to 2(4 • 2)?
(24)2
= 24 • 24
= 28
= 256
2(4 • 2)
= 28
= 256
Yes
from Rule 1
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
So, (24)2 = 2(4 • 2)
Rule 3:
(am)n = amnRule 3
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
Is (3y)2 equal to 32y2?
(3y)2
= (3y) • (3y)= 3 • y • 3 • y= 3 • 3 • y • y= 32y2
Yes
we know that multiplication is
commutative
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
So, (3y)2 = 32y2
Rule 4:
(ab)n = anbnRule 4
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
= •
=
=
YesIs equal to ?x3( )
2 x2
32
x3( )
2
x2
32
x3
x3
x • x3 • 3
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Rules of Exponentiation
So,
Rule 5:
=( )abn an
bn
=x3( )
2 x2
32
Rule 5
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Exponents
Definition:
a0 = 1
Examples:
30 = 1-65.530 = 1
10000302329178273130 = 1
for all real numbers as long as
a ≠ 0Definition
MATH1003
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
DefinitionExponents
Definition:
a-n =
Examples:
2-4 =
= 3(2-5) = 3-3 =
1an
124
32
35133
for all real numbers as long as
a ≠ 0