Operations with 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 square root, factor the number under the root into its prime factors and look for pairs. For example, $$\text{Simplify } \sqrt{36} = 6$$ because 36 = 6 × 6.

The product property states that $$\text{For any non-negative numbers } a \text{ and } b, \sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$ .

The sum property states that $$\text{You cannot combine square roots directly unless they are like terms. For example, } \sqrt{2} + \sqrt{2} = 2\sqrt{2}$$ .

To add or subtract radical expressions, combine like terms. For example, $$\text{If } 3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$$ .

Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be expressed as a simple fraction. Examples: $$\text{Rational: } \frac{1}{2}, \text{ Irrational: } \sqrt{2}$$ .

The conjugate of a binomial expression is formed by changing the sign between two terms. For example, the conjugate of $$(a + b)$$ is $$(a - b)$$ .

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator. For example, to rationalize $$\frac{1}{\text{√}2}$$ , multiply by $$\frac{\text{√}2}{\text{√}2}$$ .

The area of a rectangle is calculated by multiplying its length and width. If the dimensions are $$18\text{√}20$$ and $$12\text{√}10$$ , then the area is $$18\text{√}20 \times 12\text{√}10 = 216\text{√}200$$ .

To simplify $$\text{√}45c^2$$ , factor it as $$\text{√}(9 \times 5) \times \text{√}(c^2) = 3c\text{√}5$$ .