Operations with Radicals

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.

To simplify a square root, factor the number under the root into its prime factors, and then take out pairs of factors. 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 $$\text{You cannot combine square roots directly unless they have the same radicand. For example, } \sqrt{2} + \sqrt{2} = 2\sqrt{2}$$ .

To add or subtract radical expressions, they must have the same radicand. For example, $$\text{Combine } 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$$ .

Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be expressed as a simple fraction. Examples of irrational numbers include $$\text{√2}$$ and $$\text{π}$$ .

To multiply radical expressions, multiply the numbers outside the radicals and the numbers inside the radicals separately. For example, $$\text{Multiply } \sqrt{2} \times \sqrt{3} = \sqrt{6}$$ .

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 $$\frac{1}{\sqrt{2}}$$ , multiply by $$\frac{\sqrt{2}}{\sqrt{2}}$$ to get $$\frac{\sqrt{2}}{2}$$ .

The area of a rectangle is calculated by multiplying its length by its width. If the length is $$l$$ and the width is $$w$$ , then the area is given by $$A = l \times w$$ .