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| Rules
and Laws of Powers - Exponents |
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The
rules for the manipulation of powers - Fundamental laws of
exponents (Index laws) |
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Adding
and subtracting same powers |
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Multiplying like bases
with exponents |
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Dividing like bases
with exponents |
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Zero as an exponent |
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Power rule for exponents |
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A product raised to an exponent |
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A quotient raised to an exponent |
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Negative exponents |
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Simplifying an exponential expression |
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Scientific notation |
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| The
rules for the manipulation of exponents (or
powers) - Fundamental laws of exponents (index laws) |
| Adding
and subtracting same powers |
| Only the same powers (with same base and the exponent) can be
added or subtracted, |
|
for example a5
+ a5
+ a5
= 3a5. |
| We also say
that only like or similar terms can be added. The number that multiply the power is called
coefficient. We add or subtract the same powers by adding or subtracting their
coefficients. |
| Examples: |
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a)
2a2
+ 5a
- a2
- a
= a2
+ 4a,
b) 3a4
-
7a4
+ a4
=
(3 -
7 + 1) · a4
= -3a4 |
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| The
rules for powers (or exponents) |
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Rules |
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Examples |
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am
· an
= am
+
n, |
|
a)
a3
· a4 =
a3
+ 4,
b) 24 · 25 = 24 + 5 =
29 = 512, |
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am
¸
an =
am
-
n, |
|
a)
x5
¸
x3
= x5
-
3
= x2,
b) 0.14
¸
0.13 = 0.14
-
3 = 0.1, |
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an
· bn
= (a
· b)n, |
|
a)
24 · 34 = (2
· 3)4
= 64,
b) 45 · 0.85 = (4
· 0.8)5
= 3.25, |
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an
¸
bn
= (a ¸
b)n, |
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a)
x6
¸
y6
=
(x
¸
y)6,
b) 125 ¸
35 = (12
¸
3)5 =
45, |
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(am)n =
am
· n, |
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a)
(a3)4
= a3
·
4 = a12,
b) (45)3
= 45
·
3 = 415, |
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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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| Scientific notation |
| A
number in scientific notation is written as the product of a
number called coefficient and a power of 10. |
| While converting to
scientific notation, the decimal point of the coefficient is
placed behind the first non-zero digit. The sign of the exponent of the power
indicates, how many places the decimal point was moved to the right or to the left. |
| Examples: |
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a)
0.0000007054 = 7.054 · 10-7,
b) 5.2 · 10-4
= 0.00052 |
| |
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a)
4507000000 = 4.507 · 109,
b) 1.04 · 105 = 104000 |
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| Beginning
Algebra Contents B |
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| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
