Ms. Kennedy
Grade 9 Enriched Class Notes
Laws of Exponents
Multiplication:
Multiplying exponents with same base
For exponents with the same base, we should add the exponents:
n
m
n+m
a ·a =a
Example:
3
4
2 · 2 = 23+4 = 27 = 2·2·2·2·2·2·2 = 128
Multiplying exponents with different bases but the same exponents.
When the bases are different and the exponents of a and b are the same, we can multiply a and b first:
n
n
n
a · b = (a · b)
Example:
2
2
3 · 4 = (3·4)2 = 122 = 12·12 = 144
Multiplying exponents when the base and the exponents are different. In this
case we have to calculate each exponent and then multiply:
n
m
a ·b
Example:
2
3
3 · 4 = 9 · 64 = 576
Multiplication with Negative Exponents:
Multiplying negative exponents with the same base:
For exponents with the same base, we can add the exponents:
-n
-m
-(n+m)
n+m
a ·a
=a
=1/a
Example:
-3
-4
2 · 2 = 2-(3+4) = 2-7 = 1 / 27 = 1 / (2·2·2·2·2·2·2) = 1 / 128 =
0.0078125
Multiplying negative exponents with different bases but the same exponents:
When the bases are different and the exponents of a and b are the same, we can multiply a and b first:
-n
-n
-n
a · b = (a · b)
Example:
-2
-2
3 · 4 = (3·4)-2 = 12-2 = 1 / 122 = 1 / (12·12) = 1 / 144 =
0.0069444
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Ms. Kennedy
Grade 9 Enriched Class Notes
Multiplying negative exponents with different bases and exponents:
In this case, we have to calculate each exponent and then multiply:
-n
-m
a ·b
Example:
-2
-3
3 · 4 = (1/9) · (1/64) = 1 / 576 = 0.0017361
Fractions:
Multiplying fractions with exponents with the same base:
Multiplying fractions with exponents with same fraction base:
n
m
n+m
(a / b) · (a / b) = (a / b)
Example:
3
(4/3) · (4/3)2 = (4/3)3+2 = (4/3)5 = 45 / 35 = 4.214
Multiplying fractions with exponents with same exponent:
(a / b) n · (c / d) n = ((a / b)·(c / d)) n
Example:
3
(4/3) · (3/5)3 = ((4/3)·(3/5))3 = (4/5)3 = 0.83 = 0.8·0.8·0.8 = 0.512
Multiplying fractions with exponents with different bases and exponents:
(a / b) n · (c / d) m
Example:
3
(4/3) · (1/2)2 = 2.37 · 0.25 = 0.5925
Square Roots:
Multiplying square roots with same base:
For exponents with the same base, we can add the exponents:
n
m
(n+m)/2
(√a) · (√a) = a
Example:
2
(√5) · (√5)4 = 5(2+4)/2 = 56/2 = 53 = 125
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Ms. Kennedy
Grade 9 Enriched Class Notes
Division:
Dividing exponents with same base
For exponents with the same base, we should subtract the exponents:
n
m
n-m
a /a =a
Example:
6
3
2 / 2 = 26-3 = 23 = 2·2·2 = 8
Dividing exponents with different bases
When the bases are different and the exponents of a and b are the same, we can divide a and b first:
n
n
n
a / b = (a / b)
Example:
3
3
6 / 2 = (6/2)3 = 33 = 3·3·3 = 27
Dividing exponents with different bases and exponents.
In this case, we have to calculate each exponent and then divide:
n
m
a /b
Example:
2
3
6 / 3 = 36 / 27 = 1.333
Division with Negative Exponents:
Dividing negative exponents with the same base:
For exponents with the same base, we can subtract the exponents:
-n
-m
-n-(-m)
m-n
a /a =a
=a
Example:
-3
-5
2 / 2 = 25-3 = 22 = 2·2 = 4
Dividing negative exponents with different bases but the same exponents:
When the bases are different and the exponents of a and b are the same, we can multiply a and b first:
-n
-n
-n
n
n
a / b = (a/b) = 1 / (a/b) = (b/a)
Example:
-2
-2
3 / 4 = (4/3)2 = 1.7778
Dividing negative exponents with different bases and exponents.
When the bases and the exponents are different we have to calculate each exponent and then divide:
-n
-m
m
n
a /b =b /a
Example:
-2
-3
3 / 4 = 43 / 32 = 64 / 9 = 7.111
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Ms. Kennedy
Grade 9 Enriched Class Notes
Fractions
Dividing fractions with exponents with same base:
Dividing fractions with exponents with same fraction base:
n
m
n-m
(a / b) / (a / b) = (a / b)
Example:
3
(4/3) / (4/3)2 = (4/3)3-2 = (4/3)1 = 4/3 = 1.333
Dividing fractions with exponents with same exponent but different base:
(a / b)n / (c / d)n = ((a / b)/(c / d))n = ((a·d / b·c))n
Example:
3
(4/3) / (3/5)3 = ((4/3)/(3/5))3 = ((4·5)/(3·3))3 = (20/9)3 = 10.97
Dividing fractions with exponents with different bases and exponents:
(a / b) n / (c / d) m
Example:
3
(4/3) / (1/2)2 = 2.37 / 0.25 = 9.481
Zero exponents rule
The base b raised to the power of zero is equal to one:
0
b =1
Zero exponents examples:
Five raised to the power of zero is equal to one:
0
5 =1
Minus five raised to the power of zero is equal to one:
0
(-5) = 1
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