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| Properties of
exponents
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Properties
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|
Examples |
| 1) |
a
· a
· · · a
= an,
n factors
(exponent), a
-base, |
|
a3
= a · a
· a,
24 =
2 · 2 ·
2 ·
2
= 16, |
| 2) |
a
= a1,
or a1
=
a, |
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2 = 21,
(-3)1
= -3, |
| 3) |
a0
= 1,
an ¸
an
= 1, |
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23
¸ 23
= 23 -
3 =
20
= 1,
(ab3)0
= 1, |
| 4) |
1n
=
1, |
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15
= 1, (-1)5
= -1, |
| 5) |
(-
a)2n
= a2n,
2n
-even exponents, |
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(-
5)4 = 54
= 625, |
| 6) |
(-
a)2n
-1
= - a2n
-1,
2n-1
-odd exponents, |
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(-
2)3 = -
23
=
- 8, |
| 7) |
(-1)2n
= 1, |
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(-1)6
= 1, |
| 8) |
(-1)2n
-1
= -1, |
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(-1)7
= -1. |
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| The
rules for powers or exponents
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Rules
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|
Examples |
| 1) |
am
· an
= am
+
n, |
|
a)
a3
· a4 =
a3
+ 4,
b) 24
· 25 = 24 + 5 = 29
= 512, |
| 2) |
am
¸
an
=
am
-
n, |
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a)
x5
¸
x3
= x5
-
3
= x2,
b) 0.14
¸
0.13 = 0.14
-
3 = 0.1, |
| 3) |
an
· bn
= (a
· b)n, |
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a)
24
· 34 = (2
· 3)4
= 64,
b) 45
· 0.85 = (4
· 0.8)5
= 3.25, |
| 4) |
an
¸
bn
= (a ¸
b)n, |
|
a)
x6
¸
y6
= (x
¸
y)6,
b) 125 ¸
35 = (12
¸
3)5 =
45, |
| 5) |
 |
|
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| 6) |
(am)n
=
am
· n, |
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a)
(a3)4
= a3
·
4 = a12,
b) (45)3
= 45
·
3 = 415, |
| 7) |
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| 8) |
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| 9) |
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| Simplifying
an exponential expression |
| Use
the above rules of powers (or exponential laws) and the rules of
algebra to simplify expressions with numerical and variable
bases, as show examples: |
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| Order of operations
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|
When expressions have more than one
operation, we have to follow rules for the order of operations,
i.e., the order in which to evaluate a mathematical expression:
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| 1.
First do all operations that lie inside parentheses, if there are
parentheses inside parentheses work from the innermost parentheses
out. |
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Anything
that acts as a grouping symbol, like any expression in the
numerator or denominator of a fraction or in an exponent is also
considered grouped and should be simplified before carrying out
further operations.
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| 2.
Next, do any work with exponents or radicals.
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| 3.
Then, working from left to right, do all multiplication and
division.
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4.
And finally, working from left to right, do all addition and
subtraction.
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| Example:
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|
-
8 -
3 ·
(-
4)
+ 15
¸ (-
5) =
-
8 -
(-
12)
+
(-
3) =
-
8
+ 12
-
3
= 1
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| Example:
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-
3
+ 2
· [-
2 -
3 ·
(-
4 ¸
2
+ 1)]
=
-
3 + 2
· [-
2 -
3 ·
(-
2 +
1)] =
-
3
+ 2
· [-
2 -
3 ·
(-
1)]
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=
-
3
+ 2
· [-
2 +
3]
=
-
3
+ 2
=
-
1
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| Example:
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|
Find
the value of the expression |
 |
when x
=
-
0.5 and y
=
-
1. |
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| Solution: |
Substituting
x
=
-
0.5 and y
=
-
1 gives |
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| College
algebra contents |
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© 2004 - 2020, Nabla Ltd. All rights reserved. |
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