Answer
$x\sqrt[10] {x}$
Work Step by Step
We want to divide two radicals, but they have different indices and radicands. Let's rewrite them as exponents to see if we can simplify them. According to the Law of Exponents, we can rewrite the radical as follows:
$\sqrt[n] {x^{m}} = x^{\frac{m}{n}}$
Now we can rewrite the radical in our exercise in exponent form:
$\frac{x^{3/2}}{x^{2/5}}$
Now, we can work with the fractional exponents by converting them into equivalent fractions. The equivalent fractions should have $10$ as the denominator because it is the least common multiple for both $2$ and $5$. Let's rewrite the exponents, incorporating equivalent fractions with the same denominators:
$\frac{x^{15/10}}{x^{4/10}}$
When we divide two exponents that have the same base, we keep the base as-is and subtract the exponents:
$x^{15/10 - 4/10}$
Subtract the exponents:
$x^{11/10}$
Let's change this answer into a radical because radicals were what we started with:
$\sqrt[10] {x^{11}}$
We can still simplify this expression even more:
$\sqrt[10] {x^{(10)}x}$
We can now take the $10th$ root of $x^{10}$ and remove it from under the radical sign:
$x\sqrt[10] {x}$
