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Video: Simplifying Numerical Expressions Using the Properties of Square Roots

Kathryn Kingham

Express √3 + √12 − √75 in its simplest form.

04:02

Video Transcript

Express square root of three plus the square root of 12 minus the square root of 75 in its simplest form.

What we need to ask here to simplify these radicals is do any of the numbers inside the radical have perfect squares as their factor. When I think about factors of 12, I know there’s one and 12, two and six, and three and four. This four is a perfect square and it is a factor of 12. I can break up the number 12 inside the radical into two different factors: three times four. So instead of saying the square root of 12, I can now say the square root of three times four.

From there, we can break up the parts and say the square root of three times the square root of four. And since the square root of four equals two, we can simplify even further. Now we have the square root of three times two. What we’re saying here is that the square root of 12 is equivalent, is equal to two times the square root of three.

Now, the square root of three doesn’t have any perfect squares as its factor. It’s already in its simplest form; we can just bring that one down. Now, we can check 75 for perfect square factors. One and 75, 75 is not divisible by two. Three times 25, and we can stop there because we found a perfect square. Instead of saying the square root of 75, we can say the square root of three times 25, which we can break into two pieces: the square root of three times the square root of 25.

The square root of 25 equals five, and then we can say the square root of 75 is equal to five times the square root of three. We can’t forget to bring down that subtraction sign. So here’s what we’re left with: the square root of three plus two times the square root of three minus five times the square root of three.

To simplify even further, we can use the distributive property. Our first three has a coefficient of one. There’s only one square root of threes, and each of the other terms share a factor of the square root of three. We can undistribute this square root of three. If we take out the square root of three, we’re left with one plus two minus five; that equals negative two. And I am gonna change the order here, so I don’t have to write the parentheses.

The simplified form of that expression is negative two times the square root of three. We were able to take the square root of three plus the square root of 12 minus the square root of 75 all the way down into a simplified form of negative two times the square root of three, which is much easier of a number to work with.