Definition: A radical expression contains a root symbol, such as the square root or cube root of a number or variable expression; the number under the symbol is the radicand.
What is a radical?
The square root of 49 asks: what number squared gives 49? The cube root of 64 asks: what number cubed gives 64? In the expression with index n, the small number n tells which root is meant. When no index is shown, it is 2.
How do you simplify?
Pull out the largest perfect-square factor of the radicand. The product rule says the root of ab equals the root of a times the root of b.
Example. Simplify the square root of 180. Since 180 = 36 · 5, the root is 6 times the square root of 5. For the square root of 50x3, write 25x2 · 2x, giving 5x times the square root of 2x, for x at least 0. Like radicals can be added: 4 times root 7 plus 9 times root 7 is 13 times root 7.
What do students get wrong?
They claim the root of (a + b) equals root a plus root b. Check: the root of 9 + 16 is 5, not 3 + 4 = 7. They also leave a radical unsimplified or forget that a square root of a negative number is not real.