Question

simplify the expression by using the properties of rational exponents only.
\\(\frac{3y^{\frac{-1}{3}}}{y^{\frac{4}{3}}}\\)
\\(\frac{3y^{\frac{-1}{3}}}{y^{\frac{4}{3}}}=\\)

Explanation:

Step1: Recall exponent division rule

When dividing with the same base \(a\), \( \frac{a^m}{a^n}=a^{m - n}\). Here, the base for the variable part is \(y\), so we apply this rule to the \(y\)-terms. The coefficient \(3\) remains as is for now. So we have \(3\times\frac{y^{-\frac{1}{3}}}{y^{\frac{4}{3}}}\).

Step2: Apply the exponent rule

Using \(\frac{a^m}{a^n}=a^{m - n}\) with \(m = -\frac{1}{3}\) and \(n=\frac{4}{3}\), we get \(y^{-\frac{1}{3}-\frac{4}{3}}\). Calculating the exponent: \(-\frac{1}{3}-\frac{4}{3}=-\frac{5}{3}\). So now we have \(3y^{-\frac{5}{3}}\).

Step3: Convert negative exponent to positive

Recall that \(a^{-n}=\frac{1}{a^n}\), so \(y^{-\frac{5}{3}}=\frac{1}{y^{\frac{5}{3}}}\). Thus, the expression becomes \(\frac{3}{y^{\frac{5}{3}}}\).

Answer:

\(\boldsymbol{\frac{3}{y^{\frac{5}{3}}}}\)