Adding Radicals - Math Steps, Examples & Questions

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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}

What is adding radicals?

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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.

How to add radicals

In order to add radicals:

Simplify any unlike radicals.
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}

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.

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

Combine like radical terms by adding.

Add the whole number coefficients in front of the radicals.

$7\sqrt{5}+9\sqrt{5}+6\sqrt{5}=22\sqrt{5}$

Example 3: unlike radicals

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

Simplify any unlike radicals.

$\sqrt{2}$ cannot be simplified because there is not a perfect square factor of $2.$

$\sqrt{11}$ cannot be simplified because there is not a perfect square factor of $11.$

Combine like radical terms by adding.

In this case, the radical terms cannot be added together because they are not like radicals.

$10\sqrt{2}+4\sqrt{11}$

Example 4: sums containing non-radicals

Simplify the radical expression.

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

Simplify any unlike radicals.

There are like radicals in the expression, but there is also a radical that can be simplified.

$\sqrt{25}$ is a perfect square so you can take the square root.

$\sqrt{25}=5$ (take the principal square root when simplifying)

Combine like radical terms by adding.

The expression is now, $4+6\sqrt{2}+3\sqrt{2}+5.$

The like radical terms can be combined by adding the whole number coefficients.

$6\sqrt{2}+3\sqrt{2}=9\sqrt{2}$

The integers can also be combined by adding.

$4+5=9$

This gives $4+6\sqrt{2}+3\sqrt{2}+\sqrt{25}=9+9\sqrt{2}$ or $9\sqrt{2}+9$

Example 5: one radical needs to be simplified

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

Simplify any unlike radicals.

$\sqrt{7}$ cannot be simplified because there is not a perfect square factor of $7.$

$\sqrt{28}$ can be simplified because $4$ is the perfect square factor of $28.$

$\sqrt{28}=\sqrt{4\times7}=\sqrt{4}\times\sqrt{7}=2\times\sqrt{7}=2\sqrt{7}$

Combine like radical terms by adding.

$\sqrt{7}+\sqrt{28}$ as $\sqrt{7}+2\sqrt{7},$ the radicals are alike and can now be added together.

Add the whole number coefficients in front of the radicals.

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

$\sqrt{7}+\sqrt{28}=3\sqrt{7}$

Example 6: both radicals need to be simplified

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

Simplify any unlike radicals.

$\sqrt{8}$ can be simplified; $4$ is the largest perfect square factor of $8.$

$\sqrt{8}=\sqrt{4 \times 2}=\sqrt{4} \times \sqrt{2}=2 \times \sqrt{2}=2 \sqrt{2}$

$\sqrt{72}$ can be simplified; $36$ is the largest perfect square factor of $72.$

$\sqrt{72}=\sqrt{36\times{2}}=\sqrt{36}\times\sqrt{2}=6\times\sqrt{2}=6\sqrt{2}$

Combine like radical terms by adding.

$\sqrt{8}+\sqrt{72}$ can now be rewritten as $2 \sqrt{2}+6 \sqrt{2},$ the radicals are the same and now they can be added.

Add the whole number coefficients in front of the radicals.

$2\sqrt{2}+6\sqrt{2}=8\sqrt{2}$

$\sqrt{8}+\sqrt{72}=8\sqrt{2}$

Example 7: two radicals need to be simplified

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

Simplify any unlike radicals.

$\sqrt{75}$ can be simplified; the largest perfect square factor of $75$ is $25.$

$\sqrt{75}=\sqrt{25\times{3}}=\sqrt{25}\times\sqrt{3}=5\times\sqrt{3}=5\sqrt{3}$

$\sqrt{50}$ can be simplified; the largest perfect square factor of $50$ is $25.$

$\sqrt{50}=\sqrt{25\times{2}}=\sqrt{25}\times\sqrt{2}=5\times\sqrt{2}=5\sqrt{2}$

Combine like radical terms by adding.

$\sqrt{75}+\sqrt{50}$ can be rewritten as $5\sqrt{3}+5\sqrt{2}$

After simplifying the radical expressions, they do not have like radicals. So, leave the expression in simplified form.

$\sqrt{75}+\sqrt{50}=5\sqrt{3}+5\sqrt{2}$

Teaching tips for adding 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 radicals questions

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.

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.

$2\sqrt{3}+4\sqrt{3}$ have like radicals so they can be added. .

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

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

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}

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

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

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 $\sqrt{54}+\sqrt{24}=3\sqrt{6}+2\sqrt{6}=5\sqrt{6}$

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.

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

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

$12+5\sqrt{2}+2\sqrt{2}$ 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.

The next lessons are

Radical functions
Vectors
Quadratic equations

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