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 \\(\frac{\sqrt3{y^2}}{\sqrt{y}}\\) as a single radical? explain.

Explanation:

Step1: Rewrite radicals as exponents

$\frac{\sqrt[4]{y^3}}{\sqrt{y}} = \frac{y^{\frac{3}{4}}}{y^{\frac{1}{2}}}$

Step2: Subtract exponents (division rule)

$y^{\frac{3}{4} - \frac{1}{2}} = y^{\frac{3}{4} - \frac{2}{4}} = y^{\frac{1}{4}}$

Step3: Rewrite exponent as radical

$y^{\frac{1}{4}} = \sqrt[4]{y}$

Answer:

It is possible to write $\frac{\sqrt[4]{y^3}}{\sqrt{y}}$ as a single radical. By converting the radicals to rational exponents, applying the exponent division rule, and converting back to a radical form, the expression simplifies to $\sqrt[4]{y}$. The quotient property of radicals only restricts direct radical division without first adjusting exponents; we can use rational exponents to work around the differing indices.