Video Transcript
Express the square root of 75 plus the square root of 28 minus the square root of 27 in its simplest form.
In order to simplify these square roots, we want to ask the question, do any of the values inside the radical have square factors? Let’s start with the factors of 75, one and 75 and three and 25. Three times 25 equals 75. And 25 is a square. What about 28? One times 28, two times 14, or four times seven. Four is a square. And with 27, one times 27 or three times nine. And nine is a square.
The square root of 75 equals the square root of three times 25. The square root of 28 equals the square root of four times seven. And the square root of 27 equals the square root of three times nine.
We can then rewrite them by separating them into two different radicals multiplied together. And then, simplifying the squares, rewrite the square root of 25 as five. The square root of three times the square root of 25 is the same thing as five times the square root of three. The square root of four equals two. And bring down the square root of seven. The square root of nine equals three. And bring down the square root of three. Notice here that we had subtraction and so we kept that subtraction sign.
Five times the square root of three and negative three times the square root of three can be combined. This is because five times the square root of three is saying that we have five positive square root of threes. And negative three times the square root of three is saying we have negative three square root of threes. If we combine them, a positive and a negative cancel out. So taking five times the square root of three and subtracting three times the square root of three leaves us with two square root of threes.
We have combined like terms. But because two times the square root of seven has a different root, we can’t simplify this any further. We do notice that both of these values are being multiplied by two. And so we can take out the two and say two times the square root of three plus two times the square root of seven.
The simplest form of this expression is two times the square root of three plus the square root of seven.
