Simplifying Radicals

A radical is a symbol that represents the root of a number, most commonly the square root. For example, $$\text{the square root of } 4 \text{ is represented as } \sqrt{4} = 2$$ .

To simplify a radical means to express it in its simplest form, where no perfect square factors remain under the radical sign.

$$6$$ .

$$\text{First, factor } 75 = 25 \times 3. \ \text{Then, } \sqrt{75} = \sqrt{25 \times 3} = 5\text{ } \sqrt{3} \text{, so the simplest form is } 5\text{ } \sqrt{3}$$ .

The prime factorization of 98 is $$2 \times 7^2$$ .

$$7\text{ }\sqrt{2}$$ .

$$3\text{ }\sqrt{27} = 3 \times \sqrt{9 \times 3} = 3 \times 3 \text{ }\sqrt{3} = 9\text{ }\sqrt{3}$$ .

$$11\text{ }\sqrt{2}$$ .

$$10\text{ }\sqrt{3}$$ .

A perfect square is a number that can be expressed as the square of an integer. For example, 1, 4, 9, 16, and 25 are perfect squares.