Addition and Subtraction of Square Roots

Mathematical Operations and Square Roots

Part 1

 

In this section we will see why we can add things like $5\sqrt{2}+3\sqrt{2}$ but cannot add things like $2\sqrt{5}+2\sqrt{3}$. Later we will see how multiplication and division work when radicals (square roots and such) are involved.

Addition and Subtraction: Addition is just repeated counting. The expression $5\sqrt{2}$ means $\sqrt{2}+\sqrt{2}+\sqrt{2}+\sqrt{2}+\sqrt{2}$, and the expression $3\sqrt{2}\text{ means }\sqrt{2}+\sqrt{2}+\sqrt{2}.$ So if we add those two expressions, $5\sqrt{2}+3\sqrt{2},$ we get $8\sqrt{2}$ . Subtraction works the same way.

Consider the expression $2\sqrt{5}+2\sqrt{3}$. This means $\sqrt{5}+\sqrt{5}+\sqrt{3}+\sqrt{3}.$ The square root of five and the square root of three are different things, so the simplest we can write that sum is $2\sqrt{5}+2\sqrt{3}$.

A common way to describe when square roots can or cannot be added (or subtracted) is, “If the radicands are the same you add/subtract the number in front.” This is not a bad rule of thumb, but it treats square roots as something other than numbers.

$5\times 3+4\times 3=9\times 3$

The above statement is true. Five groups of three and four groups of three is nine groups of three.

$5\sqrt{3}+4\sqrt{3}=9\sqrt{3}$

The above statement is also true because five groups of the numbers squared that is three, plus four more groups of the same number would be nine groups of that number.

However, the following cannot be combined in such a fashion.

$3\times 8+5\times 2$

While this can be calculated, we cannot add the two terms together because the first portion is three $–$ eights and the second is five $–$ twos.

$3\sqrt{8}+5\sqrt{2}$

The same situation is happening here.

Common Mistake: The following is obviously wrong. A student learning this level of math would be highly unlikely to make such a mistake.

$7\times 2+9\times 2=16\times 4$

Seven $–$ twos and nine $–$ twos makes a total of sixteen $–$ twos, not sixteen $–$ fours. You’re adding the number of twos you have together, not the twos themselves. And yet, this is a common thing done with square roots.

$7\sqrt{2}+9\sqrt{2}=16\sqrt{4}$

This is incorrect for the same reason. The thing you are counting does not change by counting it.

Explanation: Why can you add $5\sqrt{2}+3\sqrt{2}$? Is that a violation of the order of operations (PEMDAS)? Clearly, the five and square root of two are multiplying, as are the three and the square root of two. Why does this work?

Multiplication is a short-cut for repeated addition of one particular number. Since both terms are repeatedly adding the same thing, we can combine them.

But if the things we are repeatedly adding are not the same, we cannot add them together before multiplying.

What About Something Like This: $3\sqrt{40}-9\sqrt{90}$?

Before claiming that this expression cannot be simplified you must make sure the square roots are fully simplified. It turns out that both of these can be simplified.

$3\sqrt{40}-9\sqrt{90}$

$3\cdot \sqrt{4}\cdot \sqrt{10}-9\cdot \sqrt{9}\cdot \sqrt{10}$

The dot symbol for multiplication is written here to remind us that all of these numbers are being multiplied.

$3\cdot \sqrt{4}\cdot \sqrt{10}-9\cdot \sqrt{9}\cdot \sqrt{10}$

$3\cdot 2\cdot \sqrt{10}-9\cdot 3\cdot \sqrt{10}$

$6\sqrt{10}-27\sqrt{10}$

$-21\sqrt{10}$

What About Something Like This: $\sqrt{7+7}$ versus $\sqrt{7}+\sqrt{7}.$

Notice that in the first expression there is a group, the radical symbol groups the sevens together. Since the operation is adding, this becomes:

$\sqrt{7+7}=\sqrt{14}$.

Since the square root of fourteen cannot be simplified, we are done.

The other expression becomes:

$\sqrt{7}+\sqrt{7}=2\sqrt{7}.$

 

Summary: If the radicals are the same number, the number in front just describes how many of them there are. You can combine (add/subtract) them if they are the same number. You are finished when you have combined all of the like terms together and all square roots are simplified.

 

Practice Problems: Perform the indicated operation.

$\begin{array}{l}1.\sqrt{25}-5\sqrt{5}+5\\ \\ \\ 2.\sqrt{48}+3\sqrt{3}\\ \\ \\ 3.-\sqrt{75}+8\sqrt{24}+\sqrt{75}\\ \\ \\ 4.\sqrt{200}+8\sqrt{8}-2\sqrt{32}\\ \\ \\ 5.-2\sqrt{98}+16\sqrt{2}\end{array}$