# ADDITION AND SUBTRACTION OF SURDS

Two or more surds can be added together or subtracted from each other if there are like surds.

Like surds are surds that have the same number under the square root sign. For example, $\sqrt{10}$, $3\sqrt{10}$ are like surds since they have the same number under the square root sign.

The process of adding and subtraction is very simple. You will add or subtract the whole numbers while surds are added or subtracted.

$a\sqrt{b}+c\sqrt{b}=(a+c)\sqrt{b}$

The next examples illustrate how surd are added and subtracted.

👉This post extends our series on surds👈
Example 1
Simplify $10\sqrt{7}+3\sqrt{7}$

Solution
$(10+3)\sqrt{7}$
$13\sqrt{7}$

Example 2
Simplify $13\sqrt{13}-8\sqrt{13}$

Solution
$(13-8)\sqrt{13}$
$5\sqrt{13}$

Some surd required that you simplify them so you could derive a like surd

Example 3
Simplify $2\sqrt{2}+\sqrt{18}$

Solution:
$2\sqrt{2}+\sqrt{9\times2}$
$2\sqrt{2}+\sqrt{9}\times\sqrt{2}$
$2\sqrt{2}+3\sqrt{2}$
$(2+3)\sqrt{2}$
$5\sqrt{2}$

Example 4
Write $2\sqrt{32}+\sqrt{18}-3\sqrt{8}$ as simple as possible

Solution
$2\sqrt{16\times2}+\sqrt{9\times2}-3\sqrt{4\times2}$
$2\sqrt{16}\times\sqrt{2}+\sqrt{9}\times\sqrt{2}-3\sqrt{4}\times\sqrt{2}$
$8\sqrt{2}+2\sqrt{2}-6\sqrt{2}$
$(8+3-6)\sqrt{2}$
$5\sqrt{2}$

Example 4
Write $\sqrt{24}+\sqrt{6}$ in the simplest form

Solution
$\sqrt{4\times6}+\sqrt{6}$
$\sqrt{4}\times\sqrt{6}+\sqrt{6}$
$2\sqrt{6}+\sqrt{6}$
$(2+1)\sqrt{6}$
$3\sqrt{6}$

Example 5
Simplify $\sqrt{1573}-\sqrt{325}$

Solution
$\sqrt{121\times13}-\sqrt{25\times13}$
$\sqrt{121}\times\sqrt{13}-\sqrt{25}\times\sqrt{13}$
$11\sqrt{13}-5\sqrt{13}$
$(11-5)\sqrt{13}$
$6\sqrt{13}$

Example 6
Simplify $6\sqrt{28}-10\sqrt{63}+8\sqrt{112}$

Solution
$6\sqrt{4\times7}-10\sqrt{9\times7}+8\sqrt{16\times7}$
$6\sqrt{4}\times\sqrt{7}-10\sqrt{9}\times\sqrt{7}+8\sqrt{16}\times\sqrt{7}$
$12\sqrt{7}-30\sqrt{7}+32\sqrt{7}$
$(12-30+32)\sqrt{7}$
$14\sqrt{7}$

Related post
Example 7
Simplify $2\sqrt{54}+\sqrt{150}-5\sqrt{24}$ in the simplest form

Solution
$2\sqrt{9\times6}+\sqrt{25\times6}-5\sqrt{4\times6}$
$2\sqrt{9}\times\sqrt{6}-\sqrt{25}\times\sqrt{6}+5\sqrt{4}\times\sqrt{6}$
$6\sqrt{6}+5\sqrt{6}-10\sqrt{6}$
$(6+5-10)\sqrt{6}$
$\sqrt{6}$