Adding & Subtracting Radicals

A radical is an expression that includes a root, such as a square root (√) or cube root (∛). It represents the inverse operation of exponentiation.

To simplify a radical, factor the number under the root into its prime factors, and then extract any perfect squares (or cubes) from under the radical.

Radicals are like terms if they have the same index and the same radicand (the number under the root). For example, @@ ext{ } 2 ext{ } oot{3}{8} $$and$$ ext{ } 5 ext{ } oot{3}{8}@@ are like terms.

To add radicals, first simplify each radical if possible. Then, combine like terms (radicals with the same index and radicand).

To subtract radicals, simplify each radical first. Then, subtract like terms (radicals with the same index and radicand).

First, simplify: @@2 ext{ } oot{2}{80} = 2 ext{ } oot{2}{16 imes 5} = 8 ext{ } oot{2}{5} $$and$$ -6 ext{ } oot{2}{20} = -6 ext{ } oot{2}{4 imes 5} = -12 ext{ } oot{2}{5} $$. Thus,$$ 8 ext{ } oot{2}{5} - 12 ext{ } oot{2}{5} = -4 ext{ } oot{2}{5}@@.

First, simplify: @@ oot{2}{300} = oot{2}{100 imes 3} = 10 ext{ } oot{2}{3} $$and$$ oot{2}{75} = oot{2}{25 imes 3} = 5 ext{ } oot{2}{3} $$. Thus,$$ 10 ext{ } oot{2}{3} + 5 ext{ } oot{2}{3} = 15 ext{ } oot{2}{3}@@.

First, simplify: @@3^{3} oot{2}{54} = 27 ext{ } oot{2}{2 imes 27} = 27 ext{ } oot{2}{2} - 2^{3} oot{2}{2} = -8 ext{ } oot{2}{2} + 7^{3} oot{2}{-16} = 7^{3} imes 4i = 28i $$. The final result is$$ -7^{3} oot{2}{2}@@.

No, @@10 ext{ } oot{2}{5} $$and$$ 10 ext{ } oot{2}{3}@@ are not like terms, so they cannot be combined.

First, simplify: @@5 ext{ } oot{2}{75} = 5 ext{ } oot{2}{25 imes 3} = 25 ext{ } oot{2}{3} $$and$$ 8 ext{ } oot{2}{27} = 8 ext{ } oot{2}{9 imes 3} = 24 ext{ } oot{2}{3} $$. Thus,$$ 25 ext{ } oot{2}{3} + 24 ext{ } oot{2}{3} = 49 ext{ } oot{2}{3}@@.