Expert: Scotto - 2/18/2009

Question
QUESTION: Hi

can you help me learn how to do this surd?
1/3+root 2
in words= one,divided by three plus the square root of two.

ANSWER: If that 1/3 and then add √2, which is the same as √2 + 1/3, or
1/(3+√2)?

I think you really mean the second one.

Any time there is a complex number in the denominator,
multiply the top and bottom of the fraction by the conjugate.
In this case, the conjugate is 3-√2.

So, (3-√2)/((3-√2)(3+√2)) would be just like computing (3-x)(3+x).
We know that (3-x)(3+x) is 9 - x², so (3-√2)(3+√2) is 3² - 2² = 5.

That would make the final answer (3-√2)/5.

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Now if you did mean 1/3 + √2, that would turn into (1+3√2)/√2.
To rationalize this value, multiply the top and bottom by √2.

The result is (√2+3*2)/2 = (√2+6)/2 = { by putting the 6 first on
top } 3 + √2/2, where only the √2 is divided by 2.


---------- FOLLOW-UP ----------

QUESTION: your answer is what i got when working out the question but i was confused because when i checked the answers it came up like this:(3-root2)/7   ????
is this just a misprint?

Answer
The answer is wrong with the way the question is written.
It looks like they really meant the question to be
1/(3+√2), so you would multiply the top and bottom by
(3-√2), giving (3-√2)/(3-√2)(3+√2).

If this is what was meant, the answere works out to be
(3-√2)/(3²-√2²) = (3-√2)/(9-2) =(3-√2)/7.

If the problem was thought of as
  1
------
3+√2,
but written as 1/3+√2, there is an error, for () are needed to
force addition to be done first.