Question

which expression is equivalent to ( y^{\frac{3}{2}} )? (\boldsymbol{sqrt3{y^{2}}}) (\boldsymbol{\frac{y^{3}}{2}}) (\boldsymbol{sqrt{y^{3}}}) (\boldsymbol{\frac{sqrt3{y}}{sqrt{y}}})

Explanation:

Step1: Recall the exponent - radical rule

The rule for converting a rational exponent to a radical is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (where \(n\gt0\) and \(a\) is a real number for which the expression is defined).

Step2: Apply the rule to \(y^{\frac{3}{2}}\)

Here, \(a = y\), \(m = 3\), and \(n = 2\). Using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), we substitute the values:
\(y^{\frac{3}{2}}=\sqrt[2]{y^{3}}\) (and \(\sqrt[2]{x}\) is the same as \(\sqrt{x}\)), so \(y^{\frac{3}{2}}=\sqrt{y^{3}}\)

Answer:

\(\boldsymbol{\sqrt{y^{3}}}\) (corresponding to the option \(\sqrt{y^{3}}\))