Operations with Radicals

A radical expression is an expression that includes a root, such as a square root, cube root, etc. For example, $$\text{√}x$$ or $$\text{∛}y$$ .

To simplify a square root, factor the number under the root into its prime factors and look for pairs. For example, $$\text{√}18 = \text{√}(9 \times 2) = 3\text{√}2$$ .

The product property states that $$\text{√}(a \times b) = \text{√}a \times \text{√}b$$ for any non-negative numbers a and b.

The quotient property states that $$\text{√}(\frac{a}{b}) = \frac{\text{√}a}{\text{√}b}$$ for any non-negative numbers a and b, where b ≠ 0.

To add radical expressions, they must have the same index and radicand. For example, $$3\text{√}2 + 5\text{√}2 = 8\text{√}2$$ .

Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be expressed as a simple fraction. For example, $$\text{√}2$$ is irrational.

To multiply radical expressions, multiply the coefficients and the radicands separately. For example, $$2\text{√}3 \times 3\text{√}2 = 6\text{√}(3 \times 2) = 6\text{√}6$$ .

The radical form of a number is expressed using a root symbol. For example, the radical form of 16 is $$\text{√}16$$ .

To rationalize the denominator, multiply the numerator and denominator by a suitable radical that will eliminate the radical in the denominator. For example, to rationalize $$\frac{1}{\text{√}2}$$ , multiply by $$\frac{\text{√}2}{\text{√}2}$$ to get $$\frac{\text{√}2}{2}$$ .

The area of a rectangle is found by multiplying its length and width. If the dimensions are radical expressions, such as $$\text{√}a$$ and $$\text{√}b$$ , then the area is $$\text{√}a \times \text{√}b = \text{√}(a \times b)$$ .