Operations with Radicals

Just like working with variables, you can add, subtract, multiply, and divide expressions containing radicals (like square roots). The key is understanding the rules for combining them and knowing how to simplify your final answer.

Adding and Subtracting Radicals

You can only add or subtract like radicals—radicals that have the exact same index (root type) and radicand (the number inside the root).

Think of it like combining like terms in algebra: just as $2x + 3x = 5x$, you can say $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$.

Sometimes, you need to simplify the radicals first to see if they are alike.

Example: Simplify $3\sqrt{2} + \sqrt{8}$

Since $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$, the expression becomes: $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$

Multiplying Radicals

When multiplying radicals, you multiply the numbers outside the radical together, and the numbers inside the radical together: $a\sqrt{b} \cdot c\sqrt{d} = ac\sqrt{bd}$.

If you have binomials with radicals, you can use the FOIL method, or simplify the terms inside the parentheses first to make the math much easier.

Example: Simplify $(3\sqrt{2} + \sqrt{8})(\sqrt{2} - \sqrt{18})$

First, simplify the radicals inside the parentheses: $\sqrt{8} = 2\sqrt{2}$ $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

Substitute them back into the expression: $(3\sqrt{2} + 2\sqrt{2})(\sqrt{2} - 3\sqrt{2})$

Combine the like radicals inside each parenthesis: $(5\sqrt{2})(-2\sqrt{2})$

Now multiply the outside numbers and the inside numbers: $5 \cdot (-2) \cdot \sqrt{2 \cdot 2} = -10 \cdot 2 = -20$

Dividing and Rationalizing the Denominator

In math, it is standard practice not to leave a radical in the denominator of a fraction. The process of removing it is called rationalizing the denominator.

If the denominator is a binomial (has two terms) containing a square root, you eliminate the radical by multiplying both the numerator and the denominator by its conjugate. The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$.

Example: Simplify $\frac{4}{3 + \sqrt{5}}$

The denominator is $3 + \sqrt{5}$, so its conjugate is $3 - \sqrt{5}$. Multiply the top and bottom by this conjugate:

$\frac{4}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}}$

Distribute the numerator: $4(3 - \sqrt{5}) = 12 - 4\sqrt{5}$

Multiply the denominator using the difference of squares pattern $(x+y)(x-y) = x^2 - y^2$: $(3)^2 - (\sqrt{5})^2 = 9 - 5 = 4$

Put it all together and simplify the fraction: $\frac{12 - 4\sqrt{5}}{4} = \frac{12}{4} - \frac{4\sqrt{5}}{4} = 3 - \sqrt{5}$