Introduction

When dealing with sums of square roots, we can combine any terms with the same number under the square root symbol \sqrt{\phantom 0}.

For example, consider the following expression:

2\sqrt 5 + 4\sqrt 5

Remember that 2\sqrt 5 and 4\sqrt 5 mean 2\cdot\sqrt 5 and 4\cdot\sqrt 5, respectively.

We can simplify our expression by combining the two radicals, as follows:

\begin{align*} {\color{blue}{2}}\sqrt 5 + {\color{red}{4}}\sqrt 5 &=\\[5pt] \left({\color{blue}{2}} + {\color{red}{4}} \right)\sqrt 5 &=\\[5pt] 6\sqrt 5& \end{align*}

Similarly,

\begin{align*} {\color{blue}{2}}\sqrt 7 - \sqrt 7 &=\\[5pt] {\color{blue}{2}}\sqrt 7 - {\color{red}{1}}\cdot\sqrt 7 &=\\[5pt] \left({\color{blue}{2}} - {\color{red}{1}} \right)\sqrt 7 &=\\[5pt] 1\cdot \sqrt 7 &=\\[5pt] \sqrt 7.& \end{align*}

However, we cannot combine terms when the numbers under the radical are different. For example, the expression

2\sqrt 3 + \sqrt 5

cannot be simplified because there is no way of combining \sqrt 3 and \sqrt 5.

FLAG
Example: Simplifying Square Root Expressions

Simplify 8\sqrt 3 - 5\sqrt 3 + \sqrt 5.

EXPLANATION

We can combine the two \sqrt{3} terms together, as follows:

\begin{align*} 8\sqrt3 - 5\sqrt3 + \sqrt5&= \\[5pt] (8-5)\sqrt3 + \sqrt5&= \\[5pt] 3\sqrt3 + \sqrt5& \end{align*}

FLAG
Practice: Simplifying Square Root Expressions

$4\sqrt{7} + \sqrt{7} =$

a
 $5\sqrt 7$
b
 $5 $
c
 $6\sqrt 7 $
d
 $28$
e
 $4\sqrt 7 $
Practice: Simplifying Square Root Expressions

Simplify $6\sqrt{2} - 4\sqrt{2} + 5\sqrt{3} .$

a
 $2\sqrt 2 - 5\sqrt 3$
b
 $5\sqrt 3 $
c
 $5\sqrt 2 $
d
 $2\sqrt 2 + 5\sqrt 3$
e
 $2\sqrt 2 $
The Product Rule for Radicals

Recall that the product rule for radicals states that

\sqrt{a\cdot b} = \sqrt{a}\cdot \sqrt{b}

where a and b are nonnegative numbers.

We can use the product rule for radicals to simplify some expressions. Let's see an example.

FLAG
Example: Simplifying Square Root Expressions Using the Product Rule for Radicals

Simplify the expression 3\sqrt{80} -8\sqrt{5}.

EXPLANATION

First, we use the product rule for radicals to simplify \sqrt{80}, as follows:

\begin{align*} \sqrt{80} &= \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \sqrt{5} & \end{align*}

As a result,

\begin{align*} 3\sqrt{80} -8\sqrt{5} &= \\[5pt] 3(4\sqrt{5}) -8\sqrt{5} &=\\[5pt] 12\sqrt{5}-8\sqrt{5}.& \end{align*}

Now, we can combine the two \sqrt{5} terms together:

\begin{align*} 12\sqrt{5}-8\sqrt{5} &=\\[5pt] (12-8)\sqrt{5} &=\\[5pt] 4\sqrt{5} & \end{align*}

FLAG
Practice: Simplifying Square Root Expressions Using the Product Rule for Radicals

$4\sqrt{3} - 5\sqrt{12} =$

a
 $\sqrt{3}$
b
 $2\sqrt{3}$
c
 $-2\sqrt{3}$
d
 $-6\sqrt{3}$
e
 $-\sqrt{3}$
Practice: Simplifying Square Root Expressions Using the Product Rule for Radicals

Simplify $2\sqrt{6} + \sqrt{7} - \sqrt{150}.$

a
 $-5\sqrt{6} + \sqrt{7}$
b
 $4\sqrt{7}$
c
 $-3\sqrt{6} + \sqrt{7}$
d
 $4\sqrt{6}$
e
 $\sqrt{6} + \sqrt{7}$
Simplifying Expressions With Cube Roots

We can simplify cube root expressions in the same way as with square roots.

For example, consider the following expression:

4\sqrt[3]2 - 6\sqrt[3] 2

We can simplify our expression by combining the two radicals, as follows:

\begin{align*} {\color{blue}{4}}\sqrt[3] 2 - {\color{red}{6}}\sqrt[3] 2 &=\\[5pt] \left({\color{blue}{4}} - {\color{red}{6}} \right)\sqrt[3] 2 &=\\[5pt] -2\sqrt[3] 2& \end{align*}

As with square roots, we cannot combine terms when the numbers under the radical are different. For example, the expression

3\sqrt[3] 2 + \sqrt[3] 5

cannot be simplified because there is no way of combining \sqrt[3] 2 and \sqrt[3] 5.

Finally, it is not possible to combine square root and cube root terms -- not even if the numbers under the radical symbol are the same! For example, the expression

4\sqrt 2 + \sqrt[3] 2

cannot be simplified because there is no way of combining \sqrt 2 and \sqrt[3] 2.

FLAG
Example: Simplifying Expressions Containing Cube Roots

Simplify 2\sqrt[3]{5} +5 \sqrt{10} -3 \sqrt[3]{40}.

EXPLANATION

First, we use the product rule for radicals, as follows:

\begin{align*} \sqrt[3]{40} &= \sqrt[3]{8 \cdot 5} = \sqrt[3]{8} \cdot \sqrt[3]{5} = 2 \sqrt[3]{5} & \end{align*}

As a result, we have

\begin{align*} 2\sqrt[3]{5} +5 \sqrt{10} -3 \sqrt[3]{40} &= \\[5pt] 2 \sqrt[3]{5} +5 \sqrt{10} -3(2 \sqrt[3]{5}) &=\\[5pt] 2 \sqrt[3]{5} +5 \sqrt{10} -6 \sqrt[3]{5}. \end{align*}

Now, we can combine the two \sqrt[3]{5} terms together:

\begin{align*} 2 \sqrt[3]{5} +5 \sqrt{10} -6 \sqrt[3]{5} &=\\[5pt] 5 \sqrt{10}+(2 \sqrt[3]{5} -6\sqrt[3]{5}) &=\\[5pt] 5 \sqrt{10} +(2-6) \sqrt[3]{5} &=\\[5pt] 5 \sqrt{10}-4\sqrt[3]{5} & \end{align*}

FLAG
Practice: Simplifying Expressions Containing Cube Roots

$6\sqrt[3]{11} - 2\sqrt[3]{11} =$

a
 $4\sqrt{11}$
b
 $8\sqrt[3]{11}$
c
 $-2\sqrt[3]{11}$
d
 $62\sqrt{11}$
e
 $4\sqrt[3]{11}$
Practice: Simplifying Expressions Containing Cube Roots

Simplify $\sqrt[3]{54} - \sqrt{5} + \sqrt[3]{2}.$

a
 $2\sqrt[3]{2} - \sqrt{5}$
b
 $4\sqrt[3]{2} - \sqrt{5}$
c
 $3\sqrt{5}$
d
 $3\sqrt[3]{2}$
e
 $-\sqrt[3]{2} + \sqrt{5}$
Flag Content
Did you notice an error, or do you simply believe that something could be improved? Please explain below.
SUBMIT
CANCEL