Add and Subtract Radicals

How to add and subtract radicals: 2 examples and their solutions.

Example 1

Example

Solution

Before solving this, let's see what like radicals are.

Like radicals are the terms that have the same radical.

In this expression,
3√2 and +5√2 are like radicals,
because they have the same radical √2.

By the same way,
+2√5 and -8√5 are like radicals,
because they have the same radical √5.

Only like radicals can be directly added or subtracted
by adding or subtracting their coefficients.

3√2 and +5√2 are like radicals.
So 3√2 + 5√2 = (3 + 5)√2.

+2√5 and -8√5 are like radicals.
So +2√5 - 8√5 = (2 - 8)√5.

This is just like adding and subtracting the terms of a polynomial.

3 + 5 = 8
2 - 8 = -6

So 8√2 - 6√5 is the answer.

Example 2

Example

Solution

The given radicals don't have the same radical part.
So these radicals are unlike radicals.

To add and subtract these unlike radicals,
change these unlike radicals to like radicals
by simplifying √50 and √18.

Change 50 and 18 to their prime factorizations:
50 = 5⋅10 = 5⋅5⋅2 = 2⋅52
18 = 3⋅6 = 3⋅3⋅2 = 2⋅32.

Take the squared factors, 5 and 3,
out from their square roots.

Simplify a Radical

5√2, +3√2, and -√2 are all like radicals.
So 5√2 + 3√2 - √2 = (5 + 3 - 1)√2.

5 + 3 - 1 = 7

So 7√2 is the answer.