To divide square roots, we simply divide the numbers under the root. This is called the quotient rule for radicals.
To demonstrate, let's simplify the expression by applying the quotient rule:
The quotient rule also works for cube roots. For example,
In general, the quotient rule works for any quotient of th roots. So for any we have
Watch out! We cannot use the quotient rule when we have the quotient of different radicals, such as square roots and cube roots. For example, the expression cannot be simplified because the radicals and are different.
Express as the quotient of two radicals.
Applying the quotient rule, we have
$\sqrt{\dfrac{7}{4}} =$
|
a
|
$\dfrac{\sqrt{7}}{4}$ |
|
b
|
$\dfrac{\sqrt{7}}{\sqrt{2}}$ |
|
c
|
$\dfrac{7}{\sqrt{2}}$ |
|
d
|
$\dfrac{7}{2}$ |
|
e
|
$\dfrac{\sqrt{7}}{2}$ |
$\sqrt{\dfrac{3}{5}}=$
|
a
|
$\dfrac{\sqrt{3}}{15}$ |
|
b
|
$\dfrac{3}{\sqrt{5}}$ |
|
c
|
$\dfrac{\sqrt{3}}{5}$ |
|
d
|
$\dfrac{\sqrt{3}}{\sqrt{5}}$ |
|
e
|
$\dfrac{9}{\sqrt{5}}$ |
Express as a single radical.
Using the quotient rule, we have
$\dfrac{\sqrt{18}}{\sqrt{3}} =$
|
a
|
$ 3\sqrt{2}$ |
|
b
|
$ 2\sqrt{3}$ |
|
c
|
$ \sqrt{2}$ |
|
d
|
$ \sqrt{3}$ |
|
e
|
$ \sqrt{6}$ |
$\dfrac{\sqrt{5}}{\sqrt{2}} =$
|
a
|
$ \sqrt{\dfrac{25}{8}}$ |
|
b
|
$ \sqrt{\dfrac{5}{2}}$ |
|
c
|
$ \sqrt{\dfrac{25}{2}}$ |
|
d
|
$ \sqrt{\dfrac{5}{4}}$ |
|
e
|
$ \sqrt{10}$ |
Express as a single radical.
Applying the quotient rule, we have
$\dfrac{\sqrt[3]{81}}{\sqrt[3]{3}} =$
|
a
|
$3\sqrt[3]{3}$ |
|
b
|
$\dfrac{9}{\sqrt[3]{3}}$ |
|
c
|
$9\sqrt[3]{3}$ |
|
d
|
$3$ |
|
e
|
$9$ |
$\dfrac{\sqrt[3]{5}}{\sqrt[3]{7}} =$
|
a
|
$\sqrt[3]{\dfrac{5}{7}}$ |
|
b
|
$\sqrt[3]{\dfrac{25}{49}}$ |
|
c
|
$\sqrt[3]{35}$ |
|
d
|
$\dfrac{5}{\sqrt[3]7}$ |
|
e
|
$\dfrac{\sqrt[3]5}{7}$ |