Add and Subtract Radical Expressions

Questions with Solutions for Grade 10

Grade 10 questions on how to add and subtract radicals expressions with solutions are presented.

Definition
Radical expressions are like if they have the same index and the same radicand.
Examples
equation 1 are like radicals because they have the same index (root number which is 3) and the same radicand (number under the radical which is 5.
equation 2 are not like radicals because they have different radicands 8 and 9.
equation 3 are like radicals because they have the same index (2 for square root) and the same radicand 2 x.
Add and Subtract Like Radicals
Only like radicals may be added or subtracted.
Examples
Simplify the following expressions
equation 4
Solutions to the Above Examples
The above expressions are simplified by first factoring out the like radicals and then adding/subtracting.
equation 5
More Examples
Simplify the following expressions

equation 6

Solutions to Above Examples
The above expressions are simplified by first transforming the unlike radicals to like radicals and then adding/subtracting
equation 7


equation 8


equation 9
When it is not obvious to obtain a common radicand from 2 different radicands, decompose them into prime numbers. Decompose 12 and 108 into prime factors as follows.
equation 10
We now substitute 12 and 108 by their prime factors and simplify
equation 11


equation 12


equation 13


Questions With Solutions
Simplify the following expressions
equation 14

Solutions
equation 15
3. The 3 radicands in the given expression -√ 32 - 2√ 50 + 3√ 200 are different but note that 32, 50 and 200 may be written as 2 times a number that is a perfect square as follows: 32=2 * 16, 50=2 * 25 and 100=2 * 100. Substitute in the given expression and simplify.
equation 16
equation 17
equation 18
Decompose 28 and 63 into prime factors as follows: 28=2 2 * 7 , 63=3 2 * 7 and substitute into the given expression and simplify

equation 19
equation 20
7.
equation 21
equation 22

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