4. Addition and Subtraction of Radicals

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

2a + 3a = 5a

8x2 + 2x − 3x2 = 5x2 + 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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