Simplifying Surds Quiz

Simplifying Surds

Surds are irrational numbers that cannot be simplified into whole numbers or integers. They are the square roots of numbers that cannot be expressed as a ratio of two integers. Simplifying surds involves rewriting them in their simplest form by extracting square factors from underneath the root sign.

Factors of Surds

The process of simplifying surds starts by finding the factors of the number under the root sign. In the case of surds, these factors can be either rational or irrational. For example, consider the surd (\sqrt{63}). To simplify it, you would separate it into its prime factors, which in this case are (\sqrt{3}) and (\sqrt{7}).

Laws of Surds

There are three important laws of surds that can help in simplifying surds:

1. Rewrite the surd as a product of a square number and another number: If the number under the root sign has square factors, you can rewrite the surd as a product of this square number and another number. For example, (\sqrt{9} = 3).

2. Evaluate the root of the square number: Once you have rewritten the surd as a product of a square number and another number, evaluate the root of the square number. For example, (\sqrt{9} = 3).

3. Repeat if the number under the root still has square factors: If the number under the root still has square factors, repeat the process until there are no more square factors left.

Example: Simplifying (\sqrt{63})

Let's simplify (\sqrt{63}) using the laws of surds:

1. Separate (\sqrt{63}) into its prime factors: (\sqrt{3}) and (\sqrt{7}).
2. Rewrite (\sqrt{63}) as a product of these square numbers: (\sqrt{9}) and (\sqrt{7}).
3. Evaluate the root of these square numbers: (3\sqrt{7}).

So, the simplified form of (\sqrt{63}) is (3\sqrt{7}).

Exercise

Simplify the following surds:

1. (\sqrt{32})
2. (\sqrt{40})
3. (\sqrt{625})

For each surd, follow the steps outlined above to simplify it.