Surds (Radicals): Basic Operations

Surds (Radicals) are expressions with square roots, cube roots, etc., that cannot be simplified to whole numbers. They often look like messy decimals, but they can be left as neat square root symbols. Below we will look at how we can handle them using the basic operations: addition, subtraction, multiplication, and division.

What is a surd (radical)?

A surd (radical) is an expression that includes an irrational root, like (\sqrt{2}\) or (\sqrt{3}\), which cannot be simplified to a nice, whole number.

For example:

• (\sqrt{4} = 2\) is not a surd (radical) because it simplifies to a whole number.
• (\sqrt{5}\) is a surd (radical) because it doesn’t simplify to a whole number.

 

Adding and Subtracting Surds (Radicals)

Key Rule: Only Like Surds (Radicala) Can Be Added or Subtracted

• Like surds (radicals) are surd (radical)s with the same value under the square root.
• You can add or subtract them just like you would with like terms in algebra.

Example 1: Adding Like surds (radicals)

$$2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$$

Here, both terms have 3\sqrt{3}3​, so you just add the coefficients (1 + 2) in front.

Example 2: Subtracting Like surds (radicals)

$$4\sqrt{5} - 3\sqrt{5} = \sqrt{5}$$

Since both terms contain $\sqrt{5}$, we subtract the coefficients (4 - 3) to get $1\sqrt{5}$, or simply $\sqrt{5}$

Example 3: Adding/Combining Unlike surd (radical)s

$$\sqrt{2} + \sqrt{3}$$

Since $\sqrt{2}$ and $\sqrt{3}$ are different, they cannot be combined and must be left as they are.

 

 

 

 

Multiplying surds (Radicals)

Key Rule: Multiply the Numbers Inside the Roots

When multiplying surd (radical)s, you can combine them under the same root.

Example 1: Basic Multiplication of surds (radicals)

$$\sqrt{2} \times \sqrt{3} = \sqrt{6}$$

You multiply the numbers under the square roots to get $\sqrt{6}$.

Example 2: Multiplying surds (radicals) with Coefficients

$$2\sqrt{3} \times 4\sqrt{5} = 8\sqrt{15}$$

1. Multiply the coefficients: $2 \times 4 = 8$
2. Multiply the surd (radical)s: $\sqrt{3} \times \sqrt{5} = \sqrt{15}$

So the answer is $8\sqrt{15}$

 

 

 

 

Dividing Surds (Radicals)

Key Rule: Divide the Numbers Inside the Roots

When dividing surd (radical)s, you can divide the numbers under the roots as long as they divide neatly.

Example 1: Basic Division of surds (radicals)

\[ \frac{\sqrt{12}}{\sqrt{3}} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2

1. Divide the numbers under the roots: $\frac{12}{3} = 4$
2. Simplify $\sqrt{4}$ to get $2$

Example 2: Dividing surds (radicals) with Coefficients

\[ \frac{6\sqrt{10}}{3\sqrt{2}} = 2\sqrt{5} \]

1. Divide the coefficients: $\frac{6}{3} = 2$
2. Divide the surd (radical)s: $\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{5}$

So the answer is $2\sqrt{5}$