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Dividing Radicals

In math, you can divide just about everything, including radicals!

But, let's remember, we don't put fractions or decimals under radicals (in final answers), so we only divide radicals when the number that will end up under the radical is a whole number.

When radicals are divided by other radicals, you can divide the numbers and keep the product under the radical. If you have a hard time remembering this rule, you can always test the rule with square roots that you know. 

$\dfrac{\sqrt{36}}{\sqrt{9}}=\sqrt{4}=2$

Can you prove that that works?

$\dfrac{\sqrt{36}}{\sqrt{9}}=\dfrac{6}{3}=2$

The answers are the same.  This technique works.

You can divide fractions unsimplified or simplified. Review Simplifying Radicals here. Sometimes it's the same amount of work to simplify first or second.  Other times it's easier to simplify first.

Example: 

$\dfrac{\sqrt{160}}{\sqrt{20}}=\sqrt{8}=2\sqrt{2}$

You get the same answer if you simplify first.

$\dfrac{\sqrt{160}}{\sqrt{20}}=\dfrac{4\sqrt{10}}{2\sqrt{5}}=2\sqrt{2}$


 

Example 2: (In this case, you need to start by simplifying because you can't divide the numbers under the original radicals)

$\dfrac{\sqrt{120}}{\sqrt{18}}=\dfrac{2\sqrt{30}}{3\sqrt{2}}=\dfrac{2\sqrt{15}}{3}$


 

One important thing to remember when you divide radicals is, you may not leave a radical in the denominator of a fraction. If your final answer has a radical in the denominator, you must rationalize the denominator.

How do you rationalize a denominator? Remember how you can multiple anything by 1 and not change the number (identity property)? To rationalize a denominator, you multiply the final fraction with a fraction that is equal to one which, when you multiple the denominators, will eliminate the radical in the denominator by squaring the radical in the denominator. 

Example: 

$\dfrac{\sqrt{20}}{\sqrt{3}}=\dfrac{2\sqrt{5}}{\sqrt{3}}$

This fraction cannot be divided or simplified further, but you must rationalize the denominator.

To rationalize a denominator, multiply the fraction by a fraction that equals one and will get the radical out of the bottom (hint $\sqrt{3} \times \sqrt{3}=3$!)

$\dfrac{2\sqrt{5}}{\sqrt{3}}\times \dfrac{\sqrt{3}}{\sqrt{3}}=\dfrac{2\sqrt{15}}{3}$


 

When you divide radicals that have coefficients, divide the coefficients and the radicals separately, just like you would with variables. 

Example: Remember how to divide variable expressions

$\dfrac{8x^2}{4x}=2x$

$\dfrac{12xyz}{3x^2y}=\dfrac{4z}{x}$

Example: Work the same way with radicals:

$\dfrac{4\sqrt{10}}{2\sqrt{5}}=2\sqrt{2}$

The most complicated part of dividing with radicals is the process of simplifying radicals correctly, in the process of dividing.  Follow the rules of square roots and simplifying radicals, and it's not too hard. 

Practice Problems:

  • Dividing Radicals

    1. $\sqrt{10}\div\sqrt{5}$

       

    2. $\sqrt{8}\div\sqrt{32}$

       

    3. $\sqrt{3}\div\sqrt{20}$

       

    4. $\sqrt{28}\div\sqrt{6}$

       

    5. $\sqrt{15}\div\sqrt{45}$

       

    6. $\sqrt{96}\div\sqrt{18}$

       

    7. $\sqrt{\dfrac{40}{5}}$

       

    8. $\sqrt{\dfrac{21}{7}}$

       

    9. $\sqrt{\dfrac{5}{30}}$

       

    10. $\sqrt{\dfrac{11}{88}}$

       

    11. $\sqrt{\dfrac{54}{6}}$

       

    12. $\sqrt{\dfrac{2}{15}}$

       

    13. $\dfrac{\sqrt{24}}{\sqrt{2}}$

       

    14. $\dfrac{\sqrt{3}}{\sqrt{27}}$

       

    15. $\dfrac{\sqrt{5}}{\sqrt{35}}$

       

    16. $\dfrac{\sqrt{42}}{\sqrt{3}}$

       

    17. $\dfrac{\sqrt{72}}{\sqrt{12}}$

       

    18. $\dfrac{\sqrt{14}}{\sqrt{7}}$

       

    19. $\dfrac{\sqrt{13}}{\sqrt{26}}$

       

    20. $\dfrac{\sqrt{56}}{\sqrt{6}}$

    Answer Key:

Skill:

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