Question

what is \\(\frac{x^{\frac{2}{3}} y}{x^{\frac{2}{3}} y^{\frac{1}{3}}}\\) in simplest exponential form?\
\\(\circ\\) \\(y^{\frac{10}{3}}\\)\
\\(\circ\\) \\(y^{\frac{2}{3}}\\)\
\\(\circ\\) \\(y^2\\)

Explanation:

Step1: Simplify the \(x\)-terms

For the \(x\)-terms, we have \(x^{\frac{2}{3}}\) in the numerator and \(x^{\frac{2}{3}}\) in the denominator. Using the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n}\), we get:
\(x^{\frac{2}{3}-\frac{2}{3}}=x^{0}\)
Any non - zero number to the power of 0 is 1, so \(x^{0} = 1\) (we can ignore the \(x\)-term since it is equal to 1).

Step2: Simplify the \(y\)-terms

For the \(y\)-terms, we have \(y^{1}\) (which is \(y^{\frac{3}{3}}\)) in the numerator and \(y^{\frac{1}{3}}\) in the denominator. Using the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n}\), we get:
\(y^{1-\frac{1}{3}}=y^{\frac{3 - 1}{3}}=y^{\frac{2}{3}}\)

Answer:

\(\boldsymbol{y^{\frac{2}{3}}}\) (corresponding to the option with \(y^{\frac{2}{3}}\))