RIN - Operations with Radical Expressions (Lesson)

Operations with Radical Expressions

Multiplying Radicals

An expression that has a square root radical is said to be in simplest form when there is no radicand that has a perfect square factor. An expression that has a cube root radical is said to be in simplest form when there is no radicand that has a perfect cube factor.  To multiply radicals, you multiply the coefficients, or the numbers in front of the radicals, and then multiply the values in the radicands. Lastly, you simplify the radicand if possible. 

Examples:

1.  $3\sqrt[]{2}\cdot4\sqrt[]{18}$         multiply 3 • 4 and 2 • 18

$=12\sqrt[]{36}$                   the square root of 36 equals 6

$=12\cdot6$

$=72$

 

2. $4\sqrt[]{15}\cdot2\sqrt{3}$         multiply 4 • 2 and 15 • 3

$=8\sqrt{45}$                    simplify radical 45

$=8\sqrt{5}\cdot\sqrt{9}$             the square root of 9 equal 3

$=8\sqrt{5}\cdot3$                 multiply 8 • 3 

$=24\sqrt{5}$

 

3. $4\sqrt[3]{5}\cdot3\sqrt[3]{25}$        multiply 4 • 3 and 5 • 25

$=12\sqrt[3]{125}$               the cube root of 125 equals 5

$=12\cdot5$

$=60$

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Multiplying Radicals Practice

1. $5\sqrt{3}\cdot 10\sqrt{8}$

2. $2\sqrt{15}\cdot -4\sqrt{12}$

3. $2\sqrt[3]{9}\cdot 4\sqrt[3]{15}$

4. $4\sqrt[3]{-2}\cdot \sqrt[3]{12}$

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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Adding and Subtracting Radicals

Adding and subtracting radicals is a lot different than multiplying. Radicals have to be the same in order to be added or subtracted. This means that the radicand (value under the radical) and the index (the value of the root - square or cube) must be the same. See below:

Like Radicals:

$\sqrt{3}$             $2\sqrt{3}$          $-4\sqrt{3}$

$\sqrt[3]{5}$             $4\sqrt[3]{5}$           $-\sqrt[3]{5}$

NOT Like Radicals:

$\sqrt{5}$          $\sqrt[3]{5}$                the indices are different (one is a square root and the other is a cube root) 

$\sqrt[2]{5}$             $\sqrt[2]{8}$                the radicands are different

$\sqrt[3]{6}$           $\sqrt[3]{14}$               the radicands are different

 

It's important to know that sometimes the radicals can look different - but could be the same after simplifying. 

 

$\sqrt{45}+\sqrt{20}$                       these look to be different, but let's simplify them

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

$=3\sqrt{5}+2\sqrt{5}$                   they are the same and can be added!

$=5\sqrt{5}$

 

Adding and Subtracting Examples:

1. $\sqrt[2]{24}+3\sqrt[2]{6}\:$ 

$=\:\sqrt[2]{4}\sqrt[2]{6}+3\sqrt[2]{6}$                  simplify the radicals

$=2\sqrt[2]{6}+3\sqrt[2]{6}$                       they are the same, so we can add the coefficients

$=5\sqrt[2]{6}$                                     note that the radicand remains the same! 

 

2. $\sqrt{150}-5\sqrt{96}+\sqrt{18}$

$=\sqrt{25}\sqrt{6}-5\sqrt{16}\sqrt{6}+\sqrt{9}\sqrt{2}$                  simplify the radicals

$=5\sqrt{6}-5\cdot4\sqrt{6}+3\sqrt{2}$                                

$=5\sqrt{6}-20\sqrt{6}+3\sqrt{2}$                                  combine like terms (same radicands)

$=-15\sqrt{6}+3\sqrt{2}$ 

 

3. $4\sqrt[3]{81}+2\sqrt[3]{72}-3\sqrt[3]{24}$

$=4\sqrt[3]{27}\sqrt[3]{3}+2\sqrt[3]{8}\sqrt[3]{9}-3\sqrt[3]{8}\sqrt[3]{3}$                 simplify the radicals - remember these are CUBE roots

$=4\cdot3\sqrt[3]{3}+2\cdot2\sqrt[3]{9}-3\cdot2\sqrt[3]{3}$

$=12\sqrt[3]{3}+4\sqrt[3]{9}-6\sqrt[3]{3}$                                   combine like terms (same radicands)

$=6\sqrt[3]{3}+4\sqrt[3]{9}$

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Adding and Subtracting Radicals Practice

Try these problems!

1. $5\sqrt[2]{28}+2\sqrt[2]{7}-\sqrt[2]{14}$
   
   
2. $3\sqrt[2]{32}-4\sqrt[2]{2}$
   
   
3. $5\sqrt[3]{80}-12\sqrt[3]{270}$
   
   
4. $-3\sqrt[3]{40}+6\sqrt[3]{56}-7\sqrt[3]{135}$

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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