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 radical expression, factor the number under the radical into its prime factors, and then pair the factors. Each pair can be taken out of the radical as a single number.

The square root of 48 can be simplified to $$4\text{√}3$$ .

Like radicals can be combined by adding or subtracting their coefficients. For example, $$3\text{√}2 + 2\text{√}2 = 5\text{√}2$$ .

The product of two radicals can be simplified by multiplying the numbers under the radicals together. For example, $$\text{√}a \times \text{√}b = \text{√}(a \times b)$$ .

Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions.

The principle of square roots states that for any non-negative number $$a$$ , the square root of $$a$$ is a number $$b$$ such that $$b^2 = a$$ .

To rationalize the denominator, multiply the numerator and denominator by a suitable radical that will eliminate the radical in the denominator.

The square root of a variable raised to an exponent can be simplified using the rule $$\text{√}(x^n) = x^{n/2}$$ , provided that $$n$$ is even.

The sum simplifies to $$\text{√}2 - 6\text{√}5$$ .