2.02 Adding and subtracting surds

Consider an expression such as \sqrt{16}+\sqrt{9}. If we evaluate each root we get 4+3=7, but if we try to combine the terms and evaluate the addition under one root sign, that is \sqrt{16+9}, we get \sqrt{25}=5.

So we can see that \,\sqrt{a}+\sqrt{b}\, is not equal to \sqrt{a+b}. The same thing happens to \sqrt{a}-\sqrt{b} as subtraction can be thought of as adding a negative number.

So how can we add and subtract surds?

If a and b are different values, we can not simplify the expression \sqrt{a}+\sqrt{b}.

But if they were the same number, then \,\sqrt{a}+\sqrt{a}=2 \times \sqrt{a} = 2\sqrt{a}\, as whenever we add two of the same things, we can just simplify it to 2 times the thing. This can be extended to adding or subtracting multiples of surds, such as \,5\sqrt{3}-3\sqrt{3}=2\sqrt{3}\, as 5 things take away 3 things equals 2 things.

We can add and subtract like surds:

c\sqrt{a}+d\sqrt{a}=(c+d)\sqrt{a}

c\sqrt{a}-d\sqrt{a}=(c-d)\sqrt{a}

Examples

Sometimes we are asked to add and subtract surds that do no have the same number under the root sign. If we can simplify the surd so that they have the same number under the root sign, we can then collect like surds.

For example, \sqrt{12}-\sqrt{3} seems to be impossible to simplify any further, as we have two different surds involved.

However we can simplify \sqrt{12}=\sqrt{4 \times 3} down to 2\sqrt{3} so \sqrt{12}-\sqrt{3}=2\sqrt{3}-\sqrt{3} which after collecting like surds, evaluates to \sqrt{3}.

So when dealing with these kinds of problems, make sure to simply all surds first.

Examples