explain how to simplify the square root of -28 over 3
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When we encounter the square root of a negative number, we need to use the imaginary unit i, where i equals the square root of negative one. Let's work through simplifying the square root of negative twenty-eight over three. First, we separate the negative sign by writing this as the square root of negative one times twenty-eight over three. Then we can split this into the square root of negative one times the square root of twenty-eight over three. Since the square root of negative one equals i, we get i times the square root of twenty-eight over three.
Now we apply the property that the square root of a fraction equals the square root of the numerator divided by the square root of the denominator. So i times the square root of twenty-eight over three becomes i times the square root of twenty-eight divided by the square root of three.
Next, we simplify the square root in the numerator. We factor twenty-eight to find perfect square factors. Twenty-eight equals four times seven. So the square root of twenty-eight equals the square root of four times seven, which equals the square root of four times the square root of seven. Since the square root of four is two, we get two times the square root of seven. Therefore, our expression becomes i times two square root of seven over the square root of three.
To rationalize the denominator, we multiply both the numerator and denominator by the square root of three. This gives us i times two square root of seven times square root of three, all over square root of three times square root of three. In the numerator, square root of seven times square root of three equals square root of twenty-one. In the denominator, square root of three times square root of three equals three. So we get i times two square root of twenty-one over three, which we can write as two i square root of twenty-one over three.
We have successfully simplified the square root of negative twenty-eight over three. Our final answer is two i square root of twenty-one over three. This process involved using the imaginary unit i for the negative square root, separating the fraction using square root properties, simplifying the numerator by factoring, and rationalizing the denominator to eliminate square roots from the bottom. The final simplified form is two i square root of twenty-one over three.