Video: Multiplying Square Roots of Negative Numbers

Simplify √(−10) × √(−6).

01:38

Video Transcript

Simplify the square root of negative 10 times the square root of negative six.

We’ll begin by expressing each radical in terms of 𝑖. Remember 𝑖 squared is equal to negative one. So we can say the square root of negative 10 is the same as the square root of 10𝑖 squared. And similarly, the square root of negative six is the same as the square root of six 𝑖 squared. And at this point, we can split this up. We get the square root of 10 times the square root of 𝑖 squared. And since the square root of 𝑖 squared is 𝑖, we can see that the square root of negative 10 is the same as root 10 𝑖. And similarly, the square root of negative six is root six 𝑖.

Next, we multiply these together. Multiplication is commutative. So we can rearrange this a little and say it’s equal to the square root of 10 times the square root of six which is root 60 times 𝑖 squared. And since 𝑖 squared is negative one, we see that the square root of negative 10 times the square root of negative six is negative root 60. And in fact, we need to simplify this as far as possible.

There are a number of ways to do this. We could consider 60 as a product of its prime factors. Alternatively, we find the largest factor of 60 which is also a square number. In fact, that factor is four. So this means that the square of 60 is the same as the square root of four times the square root of 15 which is equal to two root 15. And we fully simplified our expression. We get negative two root 15.

Let’s look at what would have happened had we applied the laws of radicals. We would have said that the square root of negative 10 times the square root of negative six is equal to the square of negative 10 times negative six which is equal to the square root of positive 60 or two root 15 and that’s patently different to our other solution.

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