Questions: Simplify the given expression. Write the answer with positive exponents. (2 y^(1/3))^2 / y^(1/6)

Solution Steps

To simplify the given expression, we need to follow these steps:

1. Simplify the numerator by applying the power rule \((a^m)^n = a^{m \cdot n}\).
2. Combine the exponents in the numerator.
3. Subtract the exponent in the denominator from the exponent in the numerator using the quotient rule \(a^m / a^n = a^{m-n}\).
4. Ensure the final answer has positive exponents.

First, we simplify the numerator \((2 y^{\frac{1}{3}})^{2}\).

\[ (2 y^{\frac{1}{3}})^{2} = 2^{2} \cdot (y^{\frac{1}{3}})^{2} \]

\[ = 4 \cdot y^{\frac{2}{3}} \]

Next, we combine the simplified numerator with the denominator \(\frac{4 y^{\frac{2}{3}}}{y^{\frac{1}{6}}}\).

We use the quotient rule for exponents, which states \(\frac{a^{m}}{a^{n}} = a^{m-n}\).

\[ \frac{4 y^{\frac{2}{3}}}{y^{\frac{1}{6}}} = 4 \cdot y^{\frac{2}{3} - \frac{1}{6}} \]

We need to subtract the exponents \(\frac{2}{3}\) and \(\frac{1}{6}\). To do this, we find a common denominator.

\[ \frac{2}{3} = \frac{4}{6} \]

\[ \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

So, the expression simplifies to:

\[ 4 \cdot y^{\frac{1}{2}} \]

Final Answer