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The square root of a number is a number which, when multiplied by itself, yields that original number. *Special note: Recall that two positive numbers when multiplied together equal a positive number, and two negative numbers when multiplied together equal a positive number. Let's look at some examples and it will become clear. 4 x 4 = 16 and -4 x -4 = 16 therefore the square root of 16 is +4 or -4. 6 x 6 = 36 and -6 x -6 = 36 therefore the square root of 36 is +6 or -6. Simple and easy. The square root of a positive number is both a positive and a negative number. But the square root of a negative number presents another problem. We can't multiply two identical numbers together to get a negative number because of the *Special note. So how do we evaluate the square root of, say, -4? We can't do it "directly" so we have to do it this way: sq rt -4 = (sq rt 4) x (sq rt -1) Note that we "broke down" the -4 into 4 x -1 so we'd have "room to work" the problem. We can evaluate the sq rt of 4 and get +2 and -2, and we'll write the sq rt -1 as i. The italicised "i" is used to denote sq rt -1 and permit us to find the square root of a negative number, and that "i" is for "immaginary number" here. There is no real number that can be found which is the square root of a negative number. We can only "imagine" the square root of a negative number, and the "i" lets us at least write something so we can make calculations.

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Q: What is the definition of a square root?
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What is the square root of a square that has no root?

It's not a square if it has no root. If a number is a square then, by definition, it MUST have a square root. If it did not it would not be a square.


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