Math resources Algebra Radicals

Adding radicals

Adding radicals

Here you will learn about when radical expressions can be added, how to add them, and when simplifying a radical is necessary in order to add them.

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

What is adding radicals?

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.

For example, simplify or add the radical expression:

\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.

So, radicals can only be added or subtracted (simplified) if the radicals are the same.

By simplifying radicals, variables can become alike.

The radicals are not the same, so simplify the radicals and then add them.

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}

Step-by-step guide: How to simplify radicals

What is adding radicals?

What is adding radicals?

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.

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Use this quiz to check your grade 6 – grade 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th to 8th grade algebra topics to identify areas of strength and support!

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How to add radicals

In order to add radicals:

  1. Simplify any unlike radicals.
  2. Combine like radical terms by adding.

Adding radicals examples

Example 1: like radicals, simple addition

Add the radical expression:

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

  1. Simplify any unlike radicals.

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

2Combine like radical terms by adding.

Add the whole number coefficients in front of the radicals.

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

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

Example 2: like radicals, simple addition

Add the radical expression 7\sqrt{5}+9\sqrt{5}+6\sqrt{5}.

Simplify any unlike radicals.

Combine like radical terms by adding.

Example 3: unlike radicals

Simplify the radical expression 10\sqrt{2}+4\sqrt{11}.

Simplify any unlike radicals.

Combine like radical terms by adding.

Example 4: sums containing non-radicals

Simplify the radical expression.

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

Simplify any unlike radicals.

Combine like radical terms by adding.

Example 5: one radical needs to be simplified

Simplify the radical expression \sqrt{7}+\sqrt{28}.

Simplify any unlike radicals.

Combine like radical terms by adding.

Example 6: both radicals need to be simplified

Simplify the radical expression \sqrt{8}+\sqrt{72}.

Simplify any unlike radicals.

Combine like radical terms by adding.

Example 7: two radicals need to be simplified

Simplify the radical expression \sqrt{75}+\sqrt{50}.

Simplify any unlike radicals.

Combine like radical terms by adding.

Teaching tips for adding and subtracting radicals

  • 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}.

Practice adding and subtracting radicals questions

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

6\sqrt{11}
GCSE Quiz True

6\sqrt{33}
GCSE Quiz False

7\sqrt{11}
GCSE Quiz False

6\sqrt{1331}
GCSE Quiz False

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.

2. Simplify the radical expression.

 

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

5\sqrt{15}
GCSE Quiz False

3\sqrt{5}+6\sqrt{3}
GCSE Quiz True

9\sqrt{11}
GCSE Quiz False

3\sqrt{5}+2\sqrt{3}
GCSE Quiz False

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}   have like radicals so they can be added. .

 

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

 

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

3. Simplify the radical expression.

 

\sqrt{11}+\sqrt{44}

5\sqrt{11}
GCSE Quiz False

\sqrt{55}
GCSE Quiz False

11\sqrt{5}
GCSE Quiz False

3\sqrt{11}
GCSE Quiz True

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}

 

So, now the radical expression can be added because the radicals are the same.

 

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

4. Simplify the radical expression.

 

\sqrt{54}+\sqrt{24}

5\sqrt{6}
GCSE Quiz True

\sqrt{30}
GCSE Quiz False

6\sqrt{30}
GCSE Quiz False

6\sqrt{6}
GCSE Quiz False

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}

5. Simplify the radical expression.

 

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

6\sqrt{10}\sqrt{5}
GCSE Quiz False

6\sqrt{5}+5\sqrt{2}
GCSE Quiz True

6+\sqrt{10}\sqrt{5}
GCSE Quiz False

3\sqrt{1000}
GCSE Quiz False

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.

 

The simplified answer is, 6\sqrt{5}+5\sqrt{2}

6. Simplify the radical expression.

 

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

12+5\sqrt{10}+2\sqrt{2}
GCSE Quiz False

12+7\sqrt{2}
GCSE Quiz True

\sqrt{2}(3+3\sqrt{8})
GCSE Quiz False

12-3\sqrt{2}
GCSE Quiz False

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} now has like radicals.

 

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.

Adding radicals FAQs

Can you apply other operations to radical expressions?

Yes, adding and subtracting radical expressions follow the same rules. Multiplying radical expressions and dividing radical expressions are the operations you learn soon after adding and subtracting. Visit our web page on multiplying radicals and dividing radicals to learn more about the process.

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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