How do you rationalize the denominator?

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How do you rationalize the denominator?

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Answer 1 · 2014-12-23T12:52:45

When you have a root (square root for example) in the denominator of a fraction you can "remove" it multiplying and dividing the fraction for the same quantity. The idea is to avoid an irrational number in the denominator.
Consider:
$\frac{3}{\sqrt{2}}$ $3/sqrt2$
you can remove the square root multiplying and dividing by $\sqrt{2}$ $sqrt2$ ;
$\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$ $3/sqrt2*sqrt2/sqrt2$
This operation does not change the value of your fraction because $\frac{\sqrt{2}}{\sqrt{2}} = 1$ $sqrt2/sqrt2=1$ anyway and your fraction does not change by multiplying $1$ $1$ to it.

Now you can multiply in the numerator and denominator:
$\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3 \cdot \sqrt{2}}{(\sqrt{2}) \cdot (\sqrt{2})}$ $3/sqrt2*sqrt2/sqrt2=(3*sqrt2)/((sqrt2)*(sqrt2))$ giving:
$\frac{3 \cdot \sqrt{2}}{2}$ $(3*sqrt2)/2$ you have removed the square root from the denominator! (ok it went to the nominator but this is ok).

Now a problem for you; what happens when the root is not alone???!!!
If you have:
$\frac{3}{1 + \sqrt{2}}$ $3/(1+sqrt2)$ ???
You can use the same technique but...what do you use to multiply and divide?
HINT: look at what happens if you do this:
$(1 + \sqrt{2}) \cdot (1 - \sqrt{2})$ $(1+sqrt2)*(1-sqrt2)$

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How do you rationalize the denominator?

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