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Title: The Properties Of Exponents Explained
Description: These Properties are normally introduced in the course Algebra 1
Description: These Properties are normally introduced in the course Algebra 1
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PRODUCT OF POWERS PROPERTY
2
How do you simplify 7 6
× 7
?
If you recall the way
exponents
are defined, you know that this means:
(7 × 7) × (7 × 7 × 7 × 7 × 7 × 7)
If we remove the parentheses, we have the product of eight 7s, which can be written more simply
as:
8
7
This suggests a shortcut: all we need to do is add the exponents!
2
7 6 (2 + 6) 8
× 7
= 7 = 7
In general, for all real numbers b c,
a, and
,
b
a ac a(+
× c
b )
=
To multiply two powers having the same base, add the exponents
...
ZERO EXPONENTS
Many beginning students think it's weird that anything raised to the power of zero is 1
...
0
7 1 (0 + 1) 1
× 7
= 7 = 7
1
0
We know 7
= 7
...
What number times 7 equals 7? If we try 0, we
have 0 × 7 = 7
...
In general, for all real numbers a
a ≠ 0, we have:
,
0
a
= 1
0
Note that 0
is undefined
...
)
NEGATIVE EXPONENTS
You can use the product of powers property to figure this one out also
...
2
5 2 (2 + 2) 0
× 5
= 5 = 5
2
0
2
We know 5
= 25, and we know 5
= 1
...
What number times 25
equals 1? That would be its multiplicative inverse, 1/25
...
So when you
divide
two powers with the same base, you
subtract
the exponents
...
Example:
POWER OF A PRODUCT PROPERTY
When you multiply two powers with the same
exponent
, but different bases, things go a little
differently
...
POWER OF A QUOTIENT PROPERTY
This is pretty similar to the last one
...
Suppose you have a number raised to a power,
and you multiply the whole expression by itself over and over
...
To find a power of a power, multiply the exponents
...
But what if you have
1/2
an exponent which is not an integer? What, for instance, is 9?
We can fall back again on the product of powers property to find out:
1/2
1/2
(1/2 + 1/2) 1
9 × 9 = 9
= 9
1
1/2
We know 9
= 9, so 9 =
is equivalent to
...
Similarly,
a1/3
...
To recap:
Zero Exponent Property
Negative Exponent
Property
0
a 1, (
= a
≠ 0)
Product of Powers
Property
Quotient of Powers
Property
Power of a Product
Property
Power of a Quotient
Property
Power of a Power
Property
c
() abc
ab =
Rational Exponent
Property
Title: The Properties Of Exponents Explained
Description: These Properties are normally introduced in the course Algebra 1
Description: These Properties are normally introduced in the course Algebra 1