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iGCSE (2021 Edition)

1.07 Adding and subtracting surds


We know that we can add or subtract algebraic terms, such as $-3x+5x$3x+5x, as long as the terms have the same variable part.

In this example of $-3x+5x$3x+5x each term has a variable part of $x$x, but different coefficients (which are $-3$3 and $5$5).

We can combine these like terms by adding their coefficients:


In general the variable part doesn't actually have to be a pronumeral. Can we use the same rules for adding and subtracting surds?



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

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

So how can we add and subtract surds?

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

But if they were the same number, then $\sqrt{a}+\sqrt{a}$a+a = $2\times\sqrt{a}$2×a = $2\sqrt{a}$2a as whenever you add two of the same things, you can just simplify it to $2$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}$5333=23 as $5$5 things take away $3$3 things equals $2$2 things.

So it is like collecting like terms in algebra! We can summarise this as:

Adding and subtracting surds

We can add and subtract like surds:

$c\sqrt{a}+d\sqrt{a}$ca+da = $\left(c+d\right)\sqrt{a}$(c+d)a

$c\sqrt{a}-d\sqrt{a}$cada = $\left(c-d\right)\sqrt{a}$(cd)a


Using simplified surds

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}$123 seems to be impossible to simplify any further, as we have two different surds involved.

However we can simplify $\sqrt{12}=\sqrt{4\times3}$12=4×3 down to $2\sqrt{3}$23 so $\sqrt{12}-\sqrt{3}=2\sqrt{3}-\sqrt{3}$123=233 which after collecting like surds, evaluates to $\sqrt{3}$3.

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


Practice questions

Question 1

Simplify: $10\sqrt{2}+14\sqrt{2}$102+142

Question 2

Simplify: $6\sqrt{7}+7\sqrt{5}-3\sqrt{7}+8\sqrt{5}$67+7537+85

Question 3

Simplify completely: $\sqrt{45}+\sqrt{80}$45+80



Perform simple operations with surds.

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