A surd is an exact value involving a root that cannot be simplified to a whole number. More complex expressions may involve multiplying brackets containing surds or simplifying fractions that contain surds by dividing out common factors.

 

Multiplying expressions containing surds

 

When multiplying expressions with surds, use the distributive law and then simplify.

 

For example:

$$
(\sqrt{3} + 2)(\sqrt{3} + 5)
$$

 

Multiply each term:

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

 

Simplify each part:

$$
3 + 5\sqrt{3} + 2\sqrt{3} + 10
$$

 

Combine like terms:

$$
13 + 7\sqrt{3}
$$

 

Another example:

$$
(2\sqrt{5} - 3)(\sqrt{5} + 4)
$$

 

Multiply:

$$
2\sqrt{5}\sqrt{5} + 8\sqrt{5} - 3\sqrt{5} - 12
$$

 

Simplify:

$$
10 + 5\sqrt{5} - 12
$$

$$
= 5\sqrt{5} - 2
$$

 

Simplifying fractions containing surds by dividing common factors

 

Fractions containing surds can often be simplified by dividing out a common factor from the numerator and denominator.

 

For example:

$$
\frac{6\sqrt{2}}{3\sqrt{2}}
$$

 

Both the numerator and denominator contain the factor \( 3\sqrt{2} \), so divide through:

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

$$
= 2
$$

 

Another example:

$$
\frac{10\sqrt{3}}{5\sqrt{3}}
$$

 

Divide by \( 5\sqrt{3} \):

$$
= \frac{10}{5}
$$

$$
= 2
$$

 

Fractions may also simplify partially. For example:

$$
\frac{8\sqrt{12}}{4\sqrt{3}}
$$

 

First simplify the surd:

$$
\sqrt{12} = 2\sqrt{3}
$$

 

So the fraction becomes:

$$
\frac{16\sqrt{3}}{4\sqrt{3}}
$$

 

Now divide by \( 4\sqrt{3} \):

$$
= 4
$$

 

Key points to remember

 

When multiplying expressions with surds, expand carefully and then simplify.
When simplifying fractions with surds, always look for common numerical or surd factors that can be divided out before doing anything else.

 

Keeping surd expressions simplified makes later calculations clearer and helps ensure answers remain exact.