To simplify a square root, you "take out" anything that is a "perfect square"; that is, you take out front anything that has two copies of the same factor:
sqrt[4] = sqrt[2^2] = 2
sqrt[49] = sqrt[7^2] = 7
sqrt[225] = sqrt[15^2] = 15
Note that the value of the simplified radical is positive. While either of +2 and –2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2. When you solve the equation x2 = 4, you are trying to find all possible values that might have been squared to get 4. But when you are just simplifying the expression sqrt[4], the ONLY answer is "2"; this positive result is called the "principal" root. (Other roots, such as –2, can be defined using graduate-school topics like "complex analysis" and "branch functions", but you won't need that for years, if ever.)
Sometimes the argument of a radical is not a perfect square, but it may "contain" a square amongst its factors. To simplify, you need to factor the argument and "take out" anything that is a square; you find anything you've got a pair of inside the radical, and you move it out front. To do this, you use the fact that you can switch between the multiplication of roots and the root of a multiplication. In other words, radicals can be manipulated similarly to powers:
(ab)^n = a^n b^n, and the n-th root of (ab) equals the n-th root of a times the n-th root of b
Simplify sqrt[144]
There are various ways I can approach this simplification. One would be by factoring and then taking two different square roots:
sqrt[144] = sqrt[9 * 16] = sqrt[9] sqrt[16] = 3 * 4 = 12
The square root of 144 is 12.
You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. But my steps above show how you can switch back and forth between the different formats (multiplication inside one radical, versus multiplication of two radicals) to help in the simplification process.
Simplify sqrt[24] sqrt[6]
Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical?
sqrt[24] sqrt[6] = sqrt[24 * 6] = sqrt[144] = sqrt[12 * 12] = 12
Simplify sqrt[75]
sqrt[75] = sqrt[3 * 25] = sqrt[25] sqrt[3] = 5 sqrt[3]
This answer is pronounced as "five, root three". It is proper form to put the radical at the end of the expression. Not only is "sqrt[3]5" non-standard, it is very hard to read, especially when hand-written. And write neatly, because "5 sqrt[3], or 'five times the square root of three'" is not the same as "5th-rt[3], or 'the fifth root of three'".
You don't have to factor the radicand all the way down to prime numbers when simplifying. As soon as you see a pair of factors or a perfect square, you've gone far enough.
Simplify sqrt[72]
Since 72 factors as 2×36, and since 36 is a perfect square, then:
sqrt[72] = sqrt[2 * 36] = sqrt[2 * (6 * 6)] = 6 sqrt[2]
Since there had been only one copy of the factor 2 in the factorization 2×6×6, that left-over 2 couldn't come out of the radical and had to be left behind.
Simplify sqrt[4500]
sqrt[4500] = sqrt[45 * 100] = sqrt[5 * 9 * 100] = 3 * 10 * sqrt[5] = 30 sqrt[5]
