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Hey, I have a problem: solve for exact value (simplest radical form) $-3\sqrt{27}$ , the result is $-9 \sqrt3$ . I'm in 8th grade studying for a Math placement test to take trigonometry as a freshman next year. This doesn't seem to be covered in my textbook. Can anyone explain to me what's going on here? Thanks!

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Presumably you are familiar with the rule that (for positive $a$ and $b$) $\sqrt{a\times b}=\sqrt{a}\,\,\times\sqrt{b}$ Can you see how to break up 27 into $a\times b$ for the right $a$ and $b$?

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    @Anthony K: no problem, glad to help :)2011-04-29
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I'd like to emphasize a different aspect of this question. If you punch both quantities into a calculator you'll get the same number, $-15.588457\ldots$ This is the "real" meaning of the equation at hand, $ -3\sqrt{27} = -9\sqrt{3}.$ The manipulation suggested by Zev leads you to a proof of this equality, but you should understand what it means: both expressions have the same value.

Therefore, I suggest you not only familiarize yourself with the identity $\sqrt{a\times b} = \sqrt{a} \times \sqrt{b},$ but also try to understand why it's true. Please don't think of it as a rule but as an intuitively obvious property of numbers. Of course, you can only move from the former interpretation to the latter once you've internalized the meaning of and reason behind this formula.

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    I agree with Yuval, "property" might be a better choice of words than "rule".2011-04-30