Surds

UK Secondary (7-11)

Add and subtract numerical surds

Lesson

We have already learnt that two surds can be combined into one through multiplication or division, but what about addition or subtraction? Well, let's see what happens when we try to apply the same techniques!

$\sqrt{16}+\sqrt{9}$√16+√9 = $4+3$4+3 = $7$7, while on the other hand $\sqrt{16+9}$√16+9 = $\sqrt{25}$√25 = $5$5. So we can see that $\sqrt{a}+\sqrt{b}$√`a`+√`b` is **NOT** equal to $\sqrt{a+b}$√`a`+`b`, and therefore we can't use the same techniques to combine surds. The same thing happens to $\sqrt{a}-\sqrt{b}$√`a`−√`b` as subtraction can be thought of as adding a negative number.

So how CAN we add and subtract surds? Well unfortunately, 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}$2√`a` as whenever you add two of the same things in maths, 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}$5√3−3√3 = $2\sqrt{3}$2√3 as $5$5 things take away $3$3 things equals $2$2 things. It's like collecting like terms and variables in algebra! We can summarise this as:

$c\sqrt{a}+d\sqrt{a}$`c`√`a`+`d`√`a` = $\left(c+d\right)\sqrt{a}$(`c`+`d`)√`a`

$c\sqrt{a}-d\sqrt{a}$`c`√`a`−`d`√`a` = $\left(c-d\right)\sqrt{a}$(`c`−`d`)√`a`

Sometimes even though we are required to add and subtract surds that are NOT the same number, we can simplify them to become the same number. For example $\sqrt{12}-\sqrt{3}$√12−√3 seems to be impossible to solve as we have two different surds involved! However we can simplify $\sqrt{12}$√12 down to $2\sqrt{3}$2√3 so $\sqrt{12}-\sqrt{3}$√12−√3 = $2\sqrt{3}-\sqrt{3}$2√3−√3 = $\sqrt{3}$√3.

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

Simplify: $\sqrt{10}+10\sqrt{10}$√10+10√10

Simplify: $6\sqrt{2}-8\sqrt{2}-17\sqrt{2}$6√2−8√2−17√2

Fully simplify the expression $5\sqrt{2}+26\sqrt{3}+22\sqrt{3}-8\sqrt{2}$5√2+26√3+22√3−8√2

Simplify completely: $\sqrt{243}+\sqrt{3}$√243+√3