Adding and Subtracting Square Roots

We’ve already covered the addition and subtraction of numerical square roots in some detail. The procedure for adding and subtracting square roots which may contain algebraic expressions is more or less the same:

• simplify the square root in each term of the expression
• combine terms whose square roots are identical

We will illustrate this strategy with a number of simple examples.

Example 1:

Simplify

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solution:

The square roots in all four of the terms in this expression are identical, all being just . So, we just collect the “like terms":

as the simplest form of the result.

Example 2:

Simplify

solution:

Neither of the two square roots occurring here are in simplest form. So, to start, we must simplify each term.

Since

8x ³ = 2 ³x ³ = 2 ² · 2 · x ² · x

we can write

Also

18x ⁵ = 2 · 3 ² · (x ²) ² x

and so

Thus

You could leave the final answer as this last expression, or you could still do a bit of factoring to get, as the most simplified form

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Example 3:

Simplify

solution:

This example is very similar to Example 2 above, so you should use it as a practice problem. Try to solve it yourself before looking at our solution, given below.

First, simplify the individual square roots where possible. Since

18x ² = 2 · 3 ² · x ²

we get

and since

50x 4 = 2 · 5 ² · (x ²) ²

we get

Thus

as the final answer.

Example 4:

Simplify

solution:

The last three of the square roots here can be simplified slightly:

and

Thus,

as the final result.