The square root of -9 is not possible within the realm of real numbers because there is no real number that, when multiplied by itself, yields a negative result. In the context of real numbers, the square root of a positive number produces a positive number, and the square root of zero is zero. However, when dealing with negative numbers, we must introduce the concept of imaginary numbers. The square root of -9 can be expressed as 3i, where "i" represents the imaginary unit defined as the square root of -1.
On the other hand, the expression \( 3 \sqrt{-27} \) is indeed possible by using imaginary numbers. First, we simplify \(-27\) as follows: \[ \sqrt{-27} = \sqrt{-1 \times 27} = \sqrt{-1} \times \sqrt{27} = i \times \sqrt{27}. \] Since \(\sqrt{27}\) can be further simplified to \(3\sqrt{3}\), we can write: \[ \sqrt{-27} = i \times 3\sqrt{3}. \] Therefore, when we multiply this by 3, we get: \[ 3 \sqrt{-27} = 3 \times (i \times 3\sqrt{3}) = 9\sqrt{3}i. \] In summary, while the square root of -9 is expressed as 3i in the imaginary number system, the expression \( 3\sqrt{-27} \) can also be calculated using the properties of imaginary numbers, yielding a result that incorporates both real and imaginary components.
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