We begin with the same premise as the previous section. Can we divide radicals by simply dividing the radicands?

[latex]\dfrac{\sqrt{18}}{\sqrt{2}}[/latex]

Remember our philosophy, to look for a corresponding exponent rule that does what we want! The Power of a Quotient Rule (for exponents) states that [latex]{{\left( \dfrac{a}{b} \right)}^{n}}=\dfrac{{{a}^{n}}}{{{b}^{n}}}[/latex]. Again, if you imagine that the exponent is a rational number, then you can make this rule applicable for roots as well: [latex]{{\left( \dfrac{a}{b} \right)}^{\frac{1}{n}}}=\dfrac{{{a}^{\frac{1}{n}}}}{{{b}^{\frac{1}{n}}}}.[/latex] Let’s use this rule to perform the division in the example above:

[latex]\begin{align}&\quad\frac{\sqrt{18}}{\sqrt{2}}\\ =&\quad\frac{18^{1/2}}{2^{1/2}}\\ =&\quad\left(\frac{18}{2}\right)^{1/2} && \color{blue}{\textsf{Power of a Quotient used here}}\\ =&\quad\sqrt{\frac{18}{2}}\\ =&\quad 3\end{align}[/latex]

Similar to products of radicals, when dividing radicals we divide the radicands if the radicals have the same index. It is crucial that the index on the radicals be the same when we do this or else the exponent rule would not have applied.

As with multiplication, we will start with some examples featuring integers before moving on to more complex expressions.

You may have noticed that we could have simplified the radicand in [latex]\sqrt[3]{\dfrac{640}{40}}[/latex] immediately by reducing the fraction prior to applying the Quotient Rule for Radicals. How would this affect the solving process? Let’s take another look at the same problem but simplify the radicand first.

Notice how much more straightforward the second approach was. In general, check to see if you can simplify the expression in the radicand first before applying the Quotient Rule for Radicals.

In the next video, we show more examples of simplifying a radical that contains a quotient.

As with multiplication, the main idea here is that sometimes it makes sense to divide and then simplify, and other times it makes sense to simplify and then divide. Whichever order you choose, though, you should arrive at the same final expression.

Now let us turn to some examples with variables. Notice that the process for dividing these is the same as it is for dividing integers.

In our last video, we show more examples of simplifying radicals that contain quotients with variables.

Be careful to only apply the Quotient Rule for Radicals if the index is the same. For example, while you can think of [latex]\dfrac{\sqrt{8{{y}^{2}}}}{\sqrt{225{{y}^{4}}}}[/latex] as being equivalent to [latex]\sqrt{\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[/latex] since both the numerator and the denominator are square roots, you cannot express [latex]\dfrac{\sqrt{8{{y}^{2}}}}{\sqrt[4]{225{{y}^{4}}}}[/latex] as [latex]\sqrt[4]{\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[/latex] or [latex]\sqrt{\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[/latex].

Summary

The Quotient Rule for Radicals is similar to the Product Rule and states that [latex]\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}[/latex]. Both forms (fraction bar inside or outside the radical sign) can be useful for simplifying radical expressions.