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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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up vote 3 down vote accepted

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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I believe so. It's my understanding that you use factors with the product of $a \times b$ and ideally picking a set with at least one perfect square. – Anthony K Apr 29 '11 at 22:51
@Anthony K: Exactly right! There is in fact such an ideal pair of factors for 27. Do you know which numbers go into 27? – Zev Chonoles Apr 29 '11 at 22:56
Oh my Darwin. I can't believe I forgot this! I get it now! Thanks! – Anthony K Apr 29 '11 at 22:58
@Anthony K: no problem, glad to help :) – Zev Chonoles Apr 29 '11 at 23:04

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 wholeheartedly disagree with the first assertion. Equality is much, much deeper than 'evaluating these on the calculator gives the same result', and even in 8th grade I think it's worth explaining (as you say) the why of the equation, far beyond 'they're numerically the same'. – Steven Stadnicki Apr 29 '11 at 22:52
I was worried about the "mindless manipulation" aspect of the question, especially because of Zev's choice of the term rule. – Yuval Filmus Apr 30 '11 at 0:00
I agree with Yuval, "property" might be a better choice of words than "rule". – J. M. Apr 30 '11 at 1:15

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