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

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

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

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

Continued below

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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