Question

The quotient property of radicals requires the indices of the radicals to be the same. Does this mean that it is not possible to write es002-1.jpg as a single radical? Explain.

Answer 1

The radicands are powers of the same base, so they can be written using rational exponents.

Simplify the quotient of the exponential expressions by getting a common denominator and subtracting exponents.

The simplified expression is the 4th root of y

Answer 2

Although the expression is not included, it is possible to assure that the answer is negative.

It is negative because you can express a quotient of radicals as a single radical using the least common index fo the radicals.

Look. This is the quotient property of radicals:

$\frac{ \sqrt[n]{x} }{ \sqrt[n]{y} } = \sqrt[n]{ \frac{x}{y} }$

Now, if you have the quotient of two radicals with different indices you drive the radicals to one common index, i.e.  least common index, as per this:

$\frac{ \sqrt[n]{x} }{ \sqrt[m]{x} } = \frac{ \sqrt[m.n]{x.x^m} }{ \sqrt[m.n]{y.y^n} } = \sqrt[m.n]{ \frac{x^{m+1}}{y^{n+1}} }$

As, you see it is proved that two any quotient of radicals can be rewriten using the quotient property.