Dividing Radicals
The quotient of two radicals with the same index can be written as a single radical with that index, whose radicand is the quotient of the original radicands. $$ \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{ \frac{a}{b} } $$ Here a≥0 and b>0 are real numbers with b≠0, and n>0 is a positive integer.
When dividing radicals with different indices, the expressions must first be rewritten with a common index. This is done by determining the least common multiple of the indices and expressing each radical accordingly.
A practical example
Example 1
Consider the quotient
$$ \frac{ \sqrt[3]{20} }{ \sqrt[3]{4} } $$
Because the two radicals have the same index, they may be combined into a single radical of index 3 by dividing the radicands.
$$ \frac{ \sqrt[3]{20} }{ \sqrt[3]{4} } = \sqrt[3]{ \frac{20}{4} } = \sqrt[3]{5} $$
Example 2
Now consider the quotient
$$ \frac{ \sqrt[3]{4} }{ \sqrt{5} } $$
Here the two radicals do not share the same index.
To proceed, we rewrite them using the least common index, that is, the least common multiple of 3 and 2.
$$ \mathrm{lcm}(3,2) = 6 $$
With index 6 as the common index, we apply the invariant property of radicals to express both radicals accordingly.
First, multiply the index and the exponent of the radicand in the first radical by 2.
$$ \frac{ \sqrt[3 \cdot \color{red}2]{4^{1 \cdot \color{red}2}} }{ \sqrt[2]{5} } $$
$$ \frac{ \sqrt[6]{4^2} }{ \sqrt[2]{5} } $$
Next, multiply the index and the exponent of the radicand in the second radical by 3.
$$ \frac{ \sqrt[6]{4^2} }{ \sqrt[2 \cdot \color{red}3]{5^{1 \cdot \color{red}3}} } $$
$$ \frac{ \sqrt[6]{4^2} }{ \sqrt[6]{5^3} } $$
Both radicals now have index 6.
We may therefore apply the rule for dividing radicals with equal indices.
$$ \frac{ \sqrt[6]{4^2} }{ \sqrt[6]{5^3} } = \sqrt[6]{ \frac{4^2}{5^3} } $$
This expresses the quotient as a single radical.
Proof
We now establish the rule for dividing two radicals:
$$ \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{ \frac{a}{b} } $$
Raise both sides of the equation to the nth power.
$$ \left( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \right)^n = \left( \sqrt[n]{ \frac{a}{b} } \right)^n $$
On the right-hand side, the radical and the exponent n cancel.
$$ \left( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \right)^n = \left( \sqrt[{ \not{n} }]{ \frac{a}{b} } \right)^{ \not{n} } $$
$$ \left( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \right)^n = \frac{a}{b} $$
On the left-hand side, apply the properties of exponents, specifically (a/b)x = ax / bx.
$$ \frac{( \sqrt[n]{a} )^n}{( \sqrt[n]{b} )^n} = \frac{a}{b} $$
Each radical raised to the nth power simplifies directly.
$$ \frac{( \sqrt[ \not{n} ]{a} )^{ \not{n} }}{( \sqrt[\not{n}]{b} )^{ \not{n} }} = \frac{a}{b} $$
$$ \frac{a}{b} = \frac{a}{b} $$
The identity holds, which confirms the validity of the rule for dividing radicals.
Note. An alternative argument consists in rewriting the quotient as a product: $$ \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{a} \cdot \frac{1}{\sqrt[n]{b}} $$ One then proceeds exactly as in the proof of the product rule for radicals.
And so on.
