# 4. Addition and Subtraction of Radicals

In algebra, we can combine terms that are **similar**
eg.

2

a+ 3a= 5a8

x^{2}+ 2x− 3x^{2}= 5x^{2}+ 2x

Similarly for surds, we can combine those that are similar. They must have the same radicand (number under the radical) and the same index (the root that we are taking).

### Example 1

(a) 2√7 − 5√7 + √7

Answer

In this question, the radicand (the number under the square root) is 7 in each item, and the index is 2 (that is, we are taking square root) in each item, so we can add and subtract the like terms as follows:

2√7 − 5√7 + √7 = −2√7

What I did (in my head) was to factor out √7 as follows:

2√7 − 5√7 + √7

= (2 − 5 + 1)√7

= −2√7

(b) `root(5)6+4root(5)6-2root(5)6`

Answer

Once again, each item has the same radicand (`6`) and the same index (`5`), so we can collect like terms as follows:

`root(5)6+4root(5)6-2root(5)6=3root(5)6`

(c) `sqrt5+2sqrt3-5sqrt5`

Answer

In this example, the like terms are the √5 and −√5 (same radicand, same index), so we can add them, but the √3 term has a different radicand and so we cannot do anything with it.

`sqrt5+2sqrt3-5sqrt5=2sqrt3-4sqrt5`

### Example 2

(a) `6sqrt7-sqrt28+3sqrt63`

(b) `3sqrt125-sqrt20+sqrt27`

### Example 3

Simplify:

`sqrt(2/(3a))-2sqrt(3/(2a)`

### Exercises

Q1 `sqrt7+sqrt63`

Q2 `2sqrt44-sqrt99+sqrt2sqrt88`

Q3 `root(6)sqrt2-root(12)(2^13)`

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