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Expert: - 10/27/2008


Question
QUESTION: Solve for x: 3x^4 - 22x^2 + 7 = 0

I get x = plus or minus the square root of seven

and

x = plus or minus the square root of 1/3

The book shows this to be equal to plus of minus the square root of 3 over 3. Are these two radicals really equal, or is there an error in the book?

Thank you.


ANSWER: Just going at the problem, it can readily be seen that it factors into (3x²-1)(x²-7).  The answer are ±1/√3 and ±1/√7.

Books frequently (and so do most mathematicians) rationalize the denominator.  Doing so would make the problems be
±√3/3 and ±√7/7.  This is done by multiplying the first one by
√3/√3 and the second one by √7/√7.

This was where I learned (back in school somewhere) that numbers could be written in different ways, since 1/√3 = √3/3 and 1/√7 = √7/7.

The more advanced in math you go, the more of this you will see.

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

QUESTION: The answer is actually +-√7 and +-√1/3.  My book rewrites +-√1/3 (entire fraction of 1/3 is under one large radical symbol) as +-√3/3 (Numerator only under the radical. Is there a principle or property of radicals you are aware of that would make me adjust my answer? The author threw me a curve by putting that in there with no explanation.

Answer
√3/3 is the same as 1/√3 .  All that has to be done is to multiply by √3/√3 and you'll get the their answer.  This is because almost all people in math like to rationalize the denominator and I forgot to do so to 1/√3.  You're right (or, the books right), it really is √3/3.

Part of that is due to the fact that we like to know if we're talking halves (/2), thirds (/3), fourths (/4), 17ths (/17) or whatever.  If two number come up, it is much easier to compare them when there are two fractions involved.  It is also easier to think about them.  When someone says they have 3/4 of something, that's much easier to understand than someone who says,
'Yeah, I got 7/(5√2) - hey - that's almost 1.'

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