Students first learn how to add radicals in algebra and build upon those strategies as they progress through high school math.

Adding radicals is where you can add radical expressions when the numbers or expressions under the root symbols (the radicands) are the same; these are called ‘like radicals’. The strategy of adding like roots and radicals is very similar to the strategy of adding like algebraic terms (combining like terms).

Recall simplifying a+a+2a\text{:}

Gather like terms by adding them together.

a+a+2a=1a+1a+2a=4a

This strategy can be applied to radicals.

\sqrt{3}+\sqrt{3}+2\sqrt{3}

If this was x+x+2x, you would gather these terms to get 4x.

As \sqrt{3}=1\sqrt{3}, \sqrt{3}+\sqrt{3}+2\sqrt{3} is 1\sqrt{3}+1\sqrt{3}+2\sqrt{3}

Adding or combining the like radicals simplifies the expression to be: 4\sqrt{3}. Notice how the number in front of the like radicals is what you are adding, not the number under the radical (exactly the same if you were gathering algebraic terms).

\sqrt{3}+\sqrt{3}+2 \sqrt{3}=4 \sqrt{3}

If the radicals are not the same, you cannot simplify the expression, just like when the variables of an algebraic expression are not the same, you cannot simplify it.

• 2a+5b cannot be simplified because the variables are not alike.
• 2\sqrt{3}+5\sqrt{7} cannot be simplified because the radicals are not alike.

By simplifying radicals, variables can become alike.

For example,

\begin{aligned}&\sqrt{8}+\sqrt{32}+\sqrt{2} \\\\ &\sqrt{8}=\sqrt{4\times{2}}=\sqrt{4}\times\sqrt{2}=2\sqrt{2} \\\\ &\sqrt{32}=\sqrt{16\times{2}}=\sqrt{16}\times\sqrt{2}=4\sqrt{2} \end{aligned}

Rewrite the expression in its simplest form.

\sqrt{8}+\sqrt{32}+\sqrt{2}=2 \sqrt{2}+4 \sqrt{2}+\sqrt{2}=7 \sqrt{2}

## Common Core State Standards

How does this relate to high school math?

• High school – The Real Number System (HSN-RN.B.3)
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a non-zero rational number and an irrational number is irrational.

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

Both radicals are the same so you don’t need to simplify any radicals.

2\sqrt{3}+5\sqrt{3}=7\sqrt{3}

Remember to only add the numbers that sit in front of the radicals.

### Example 4: sums containing non-radicals

4+6\sqrt{2}+3\sqrt{2}+\sqrt{25}

### Example 7: two radicals need to be simplified

• Remind students that there is always a 1 in front of the radical symbol.

• Encourage students to use math concepts such as gathering like terms.

### Easy mistakes to make

• Thinking that a root has no integer coefficient
\sqrt{10} {0\sqrt{10}}
\sqrt{10}=1\sqrt{10}

• Forgetting to simplify each radical fully
For example, when adding 5\sqrt{8}+4\sqrt{32} simplifying the radical expression to be 5\times\sqrt{4\times{2}}+4\times\sqrt{4\times{8}}=5\times{2}\sqrt{2}+4\times{2}\sqrt{8}=10\sqrt{2}+8\sqrt{8} instead of fully simplifying using the largest square factors of 4\sqrt{32}=4\times\sqrt{16\times{2}}=4\times{4}\sqrt{2}=16\sqrt{2}, which you then can add 10\sqrt{2}+16\sqrt{2}=26\sqrt{2}.

• Trying to combine unlike radicals
It’s OK to leave an answer with more than one radical in it if it will not simplify further. For example, 7\sqrt{3}+4\sqrt{5} cannot be added together because the radicals are not the same. So, leave the most simplified answer as 7\sqrt{3}+4\sqrt{5}.

1. Simplify the expression 3\sqrt{11}+2\sqrt{11}+\sqrt{11}

6\sqrt{11}

6\sqrt{33}

7\sqrt{11}

6\sqrt{1331}

The expression 3\sqrt{11}+2\sqrt{11}+\sqrt{11} has like radicals so you can simply add left to right.

3\sqrt{11}+2\sqrt{11}+\sqrt{11}=6\sqrt{11}

Remember that \sqrt{11} has a 1 in front of it.

3\sqrt{5}+2\sqrt{3}+4\sqrt{3}

5\sqrt{15}

3\sqrt{5}+6\sqrt{3}

9\sqrt{11}

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

In the expression, 3\sqrt{5}+2\sqrt{3}+4\sqrt{3}, there are two radicals that are the same and one that isn’t.

3\sqrt{5} cannot be simplified so leave it as is.

2\sqrt{3}+4\sqrt{3}=6\sqrt{3}

So the simplified answer is: 3\sqrt{5}+6\sqrt{3}

\sqrt{11}+\sqrt{44}

5\sqrt{11}

\sqrt{55}

11\sqrt{5}

3\sqrt{11}

The expression \sqrt{11}+\sqrt{44}   does not have like radicals but \sqrt{44} can be simplified.

\begin{aligned}\sqrt{44}&=\sqrt{4\times{11}} \\\\ &=\sqrt{4}\times\sqrt{11} \\\\ &=2\times\sqrt{11} \\\\ &=2\sqrt{11} \end{aligned}

\sqrt{11}+2\sqrt{11}=3\sqrt{11}

\sqrt{54}+\sqrt{24}

5\sqrt{6}

\sqrt{30}

6\sqrt{30}

6\sqrt{6}

The expression cannot be subtracted the way it is written because the radicals are not the same. But, they both can be simplified.

\sqrt{54} can be simplified:

\begin{aligned}\sqrt{54}&=\sqrt{9\times{6}} \\\\ &=\sqrt{9}\times\sqrt{6} \\\\ &=3\times\sqrt{6} \\\\ &=3\sqrt{6}\end{aligned}

\sqrt{24} can be simplified:

\begin{aligned}\sqrt{24}&=\sqrt{4\times{6}} \\\\ &=\sqrt{4}\times\sqrt{6} \\\\ &=2\times\sqrt{6} \\\\ &=2\sqrt{6} \end{aligned}

So the expression \sqrt{54}+\sqrt{24}=3\sqrt{6}+2\sqrt{6}=5\sqrt{6}

3\sqrt{20}+\sqrt{50}

6\sqrt{10}\sqrt{5}

6\sqrt{5}+5\sqrt{2}

6+\sqrt{10}\sqrt{5}

3\sqrt{1000}

The expression, 3\sqrt{20}+\sqrt{50} does not have like radicals, but they can be simplified.

3\sqrt{20} simplifies to be:

\begin{aligned}3\sqrt{20}&=3\times\sqrt{4\times{5}} \\\\ &=3\times\sqrt{4}\times\sqrt{5} \\\\ &=3\times{2}\times\sqrt{5} \\\\ &=6\times\sqrt{5} \\\\ &=6\sqrt{5} \end{aligned}

\sqrt{50} simplifies to be:

\begin{aligned}\sqrt{50}&=\sqrt{25}\times{2} \\\\ &=\sqrt{25}\times\sqrt{2} \\\\ &=5\sqrt{2}\end{aligned}

So, 3\sqrt{20}+\sqrt{50}=6\sqrt{5}+5\sqrt{2} which still does not have like radicals.

3\sqrt{16}+\sqrt{50}+\sqrt{8}

12+5\sqrt{10}+2\sqrt{2}

12+7\sqrt{2}

\sqrt{2}(3+3\sqrt{8})

12-3\sqrt{2}

The expression, 3\sqrt{16}+\sqrt{50}+\sqrt{8} does not have like radicals, but they can be simplified.

3 \sqrt{16} simplifies to be:

\begin{aligned}3\sqrt{16}&=3\times{4} \\\\ &=12 \end{aligned}

\sqrt{50} simplifies to be:

\begin{aligned}\sqrt{50}&=\sqrt{25}\times{2} \\\\ &=\sqrt{25}\times\sqrt{2} \\\\ &=5\sqrt{2} \end{aligned}

\sqrt{8} simplifies to be:

\begin{aligned}\sqrt{8}&=\sqrt{4\times{2}} \\\\ &=\sqrt{4}\times\sqrt{2} \\\\ &=2\times\sqrt{2} \\\\ &=2\sqrt{2}\end{aligned}

So, 3\sqrt{16}+\sqrt{50}+\sqrt{8}=12+5\sqrt{2}+2\sqrt{2}

12+5\sqrt{2}+2\sqrt{2}=12+7\sqrt{2}

12+7\sqrt{2} is the most simplified answer. 12 cannot be added to 7\sqrt{2} because 12 is an integer, not a radical.

Can you apply other operations to radical expressions?

Can you add cube root expressions?

Yes, you can add cube root expressions using a similar strategy when you add square root expressions.

Is the process for solving radical equations the same as for solving quadratic equations?

The process is different. To solve simple radical equations, you need to isolate the radical part to be on one side of the equation. Then apply the correct exponent to both sides of the equation to get rid of the radical sign. For example, if the root is a cube root, then the exponent or power each side of the equation needs to be raised to is 3.

Can a rational expression contain a radical expression?

A rational expression can have a radical expression in the numerator or the denominator, although the value is irrational.

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