Question 74896: Can anyone help me with this?
My problem is how to rationalize the denominator:
2/ sqrt[6] - sqrt[5] Found 2 solutions by ptaylor, bucky:Answer by ptaylor(2198) (Show Source):
First, multiply both numerator and denominator by we can do this because equals 1 and we want to do this to get rid of the radicals in the denominator.
and this equals
You can put this solution on YOUR website!
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I'm going to assume that both the square root terms are in the denominator as shown above.
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You can rationalize the denominator through using the identity:
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You can multiply out the left side of this identity just to convince yourself that the result
equals the right side.
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The denominator of the problem is of the form (a - b). Suppose that we multiply the entire
term of the problem by which is equivalent to multiplying the term
in the problem by 1 since the numerator of this multiplier equals the denominator.
I used "a" and "b" so it might be easier to see what we're trying to do. Actually, we're
going to let:
. and .
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So when we multiply the original problem by we're actually going
to multiply it by .
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Let's do it:
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In accordance with the identity above, after multiplying the original denominator
of the problem by the new, rationalized denominator becomes
the difference between the squares of the two terms that were in the original denominator:
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But and so the new, rationalized denominator
just equals 6 - 5 or 1.
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Now all we have left to do is to work on the numerator.
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The numerator of the original problem is 2, but we multiplied it by
as part of the process associated with rationalizing the denominator. When we do this we get:
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That's the answer ... because the new denominator that it is over
is just 1.
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Hope that after looking through this you'll be familiar with using the identity to rationalize
denominators. It will be useful to you when you work with complex numbers.
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