Simplifying a Radical with Addition?

Question

Why is addition wrong?

Simplify: $\sqrt{18}$

the correct answer: $\sqrt{9 \times 2} = 3\sqrt2$

the wrong answer: $\sqrt{16 + 2} = 4\sqrt2$

Answer 1

When you square root something, you want to find a factor of the number in the root and (ideally) you will find a perfect square.

You cannot use addition in square roots because, if you think about it, there can be a multitude of answers, when in reality, there is only one, true answer.

For example, if you tried to find the square root of 9+9, which still equals 18, your answer would be very different.

Additionally, the following logic must be applied:

(a+b)(a+b) = a^2 + 2ab + b^2 which is not a^2+b^2

(sorry about the symbols, I am having difficulty implementing those)

Answer 2

If $\sqrt{a+b}=\sqrt a \sqrt b$,

then (squaring both sides) $ a+b=ab,$

which does not generally hold.