Mixed 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 radicals, they must have the same index and radicand. For example, $$\text{Combine } 3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$$ .

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 area of a rectangle is calculated by multiplying its length by its width. Formula: $$\text{Area} = \text{length} \times \text{width}$$ .

First, simplify $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ . Then, $$18\sqrt{20} = 18 \times 2\sqrt{5} = 36\sqrt{5}$$ .

If a rectangle has dimensions of $$a\sqrt{b}$$ and $$c\sqrt{d}$$ , then the area is $$\text{Area} = (a\sqrt{b}) \times (c\sqrt{d}) = ac\sqrt{bd}$$ .

To multiply radicals, multiply the numbers outside the radicals and the numbers inside the radicals separately. For example, $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$ .