Operations with Radicals

Add and Subtract Radicals

One reason to learn how to simplify radicals is to be able to combine them through addition and subtraction. Radicals are called similar if they have the same index and same radicand, such as $4\cdot\sqrt[3]{xy}$ and $7\cdot\sqrt[3]{xy}$. Taking the sum of these two terms would result to $11\cdot\sqrt[3]{xy}$. Notice how only the coefficients $4$ and $7$ are combined, while everything else remains the same.

Multiplying Radicals

To multiply two or more radicals together, make sure that the index of each factor match. When the indices match, the process is relatively easy: write all the factors underneath one common radical, then simplify the radicand.

On the contrary, when they’re different, each factor needs to be algebraically manipulated to match before being multiplied. Both technique are fully demonstrated in the video below.

Dividing Radicals

To be successful at dividing radicals, you’ll need to remember an important skill involving how to rationalize the denominator. In addition, you’ll need to remember how to divide a polynomial by a monomial (watch here). A visual demonstration on how to divide radicals is provided below:

Conjugate Method

When dividing by a binomial containing square roots, multiply the divisor (denominator) and the dividend (numerator) by the conjugate of that binomial. The conjugate of a binomial is a binomial having the same two terms, but differ only in the sign of one term. Thus, the conjugate of $a+b$ is $a-b$. Recall that when we multiply a binomial, say, $(a+b)$ by its conjugate $(a-b)$, we get:

$$(a+b)(a-b) \Rightarrow a^2 – ab + ab – b^2$$

The two middle terms can cancel out, leaving us with a difference of two squares: $a^2-b^2$. Therefore, if $a$ and $b$ were square roots, our final expression would have no square roots.