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 $\frac{sqrt4{y^{2}}}{sqrt{y}}$ as a single radical? explain.

Explanation:

Step1: Convert radicals to exponents

Use the rule \(\sqrt[n]{a}=a^{\frac{1}{n}}\). So, \(\sqrt[4]{y^{2}} = y^{\frac{2}{4}}=y^{\frac{1}{2}}\) and \(\sqrt{y}=y^{\frac{1}{2}}\).

Step2: Apply the quotient rule for exponents

The quotient rule \(a^{m}\div a^{n}=a^{m - n}\). Here, \(\frac{y^{\frac{1}{2}}}{y^{\frac{1}{2}}}=y^{\frac{1}{2}-\frac{1}{2}}=y^{0}\). Since \(y^{0} = 1=\sqrt[4]{1}\) (or any radical of \(1\) since \(1^{n}=1\) for any \(n>0\)). Another way: rewrite with a common index. The least - common multiple of \(2\) and \(4\) is \(4\). \(\sqrt{y}=\sqrt[4]{y^{2}}\). Then \(\frac{\sqrt[4]{y^{2}}}{\sqrt[4]{y^{2}}}=\sqrt[4]{\frac{y^{2}}{y^{2}}}=\sqrt[4]{1}\)

Answer:

It is possible. We can rewrite the radicals with a common index (using the property \(\sqrt[n]{a}=a^{\frac{1}{n}}\) and the quotient rule for exponents \(a^{m}\div a^{n}=a^{m - n}\)). For example, \(\frac{\sqrt[4]{y^{2}}}{\sqrt{y}}\), rewrite \(\sqrt{y}=\sqrt[4]{y^{2}}\), then \(\frac{\sqrt[4]{y^{2}}}{\sqrt[4]{y^{2}}}=\sqrt[4]{\frac{y^{2}}{y^{2}}}=\sqrt[4]{1}\) (or using the exponent form: \(\sqrt[4]{y^{2}}=y^{\frac{2}{4}} = y^{\frac{1}{2}}\), \(\sqrt{y}=y^{\frac{1}{2}}\), \(\frac{y^{\frac{1}{2}}}{y^{\frac{1}{2}}}=1=\sqrt[4]{1}\))