Dividing with Indices
On page two we looked at why you add the indices when multiplying
two powers with the same base. Now lets look at the rule for dividing.
Click on the parts of the equation to find the answer.
When we divide powers with the same base we subtract the indices.
You know by now that it is important that we understand why this
rule works. It is a simple rule and not so hard to explain, but
the explanations involve many rules of multiplication, division
and fractions. Here are three different explanations that will give
you an idea of some of the issues that
are involved.
Explanation 1: Using the rule for multiplying.
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By the rule for muliplying
powers with the same base by adding the indices. The addition
undoes or cancels the subtraction. |
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Dividing both sides of the
equation by . (Click on the equation
to cancel) |
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On the RHS (right hand side)
the division undoes or cancels the multiplication. |
Explanation 2: Using Fractions
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This is an important relationship
between division and fractions. |
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Using the definitions.
Notice that we used the alternative definition in the denominator.
This will make it clearer when we cancel. |
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Here we have cancelled
common factors in the numerator and denominator. Click to
make it happen. |
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After cancelling five
of the seven factors in the numerator there are only left. |
Explanation 3: Undoing the multiplications one at a time.
Consider the expression
Each of the divisions cancels out each of the multiplications,
one at a time. Click to see it happen, starting in the middle. When
all five multiplications have been cancelled there are only factors left. But dividing by three five times
is the same as dividing by
. This is why:
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Dividing by a number is the same as multiplying
by its reciprocal |
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Regrouping. This means a different order
of operartions. |
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The rule for multiplying fractions. |
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Definition of indices. |
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Multiplying by a number is the same as dividing
by its reciprocal |
Thus we can conclude that and are the same thing.
Here is a formal statement of the rule for subtracting indices.
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Positive
Indices
