radical operations

A radical is an expression that includes a root, such as a square root (√) or cube root (∛). It represents the operation of finding a number that, when multiplied by itself a certain number of times, gives the original number.

The square root of a number x is a value y such that $$y^2 = x$$ . It is denoted as $$\text{√}x$$ .

To simplify a radical expression, factor the number under the radical into its prime factors, and then take out pairs of factors for square roots (or appropriate groups for higher roots).

The product of two square roots can be simplified using the property: $$\text{√}a \times \text{√}b = \text{√}(a \times b)$$ .

The sum of two radicals can only be simplified if they have the same radicand (the number under the radical). For example, $$\text{√}a + \text{√}a = 2\text{√}a$$ .

The difference of two squares is a special product that can be factored as: $$a^2 - b^2 = (a + b)(a - b)$$ .

To multiply a radical by a whole number, simply multiply the whole number by the coefficient of the radical. For example, $$3 \times \text{√}2 = 3\text{√}2$$ .

The simplified form is $$20x\text{√}3$$ .

The result is $$15\text{√}2$$ .

$$\text{√}15$$ cannot be simplified.