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{\sqrt4{y^3}}{\sqrt{y}}\\) as a single radical? explain.

Explanation:

Step1: Rewrite radicals as exponents

The fourth - root of \(y^{3}\) can be written as \(y^{\frac{3}{4}}\), and the square - root of \(y\) can be written as \(y^{\frac{1}{2}}\). So, \(\frac{\sqrt[4]{y^{3}}}{\sqrt{y}}=\frac{y^{\frac{3}{4}}}{y^{\frac{1}{2}}}\).

Step2: Use the quotient rule of exponents

According to the rule \(a^{m}\div a^{n}=a^{m - n}\), when \(a = y\), \(m=\frac{3}{4}\), and \(n=\frac{1}{2}\), we have \(y^{\frac{3}{4}-\frac{1}{2}}\).

Step3: Simplify the exponent

\(\frac{3}{4}-\frac{1}{2}=\frac{3 - 2}{4}=\frac{1}{4}\). So, \(y^{\frac{3}{4}-\frac{1}{2}}=y^{\frac{1}{4}}\).

Step4: Rewrite the exponent as a radical

By the definition \(a^{\frac{1}{n}}=\sqrt[n]{a}\), when \(a = y\) and \(n = 4\), \(y^{\frac{1}{4}}=\sqrt[4]{y}\).

Answer:

It is possible. \(\frac{\sqrt[4]{y^{3}}}{\sqrt{y}}=\sqrt[4]{y}\)