Question

Which expression is equivalent to (x3y/2)4
?(1 point)
Responses

x12y6
the fraction with numerator x to the 12th power and denominator y to the 6th power

x7y8
the fraction with numerator x to the 7th power and denominator y to the 8th power

x12y8
the fraction with numerator x to the 12th power and denominator y to the 8th power

x7y6

Answer 1

To simplify the expression \((\frac{x^3y}{2})^4\), we will apply the power of a quotient rule, which states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\), along with the power of a product rule, which states that \((ab)^n = a^n b^n\).

1. Start by applying the exponent to the numerator and the denominator:

\[
\left(\frac{x^3y}{2}\right)^4 = \frac{(x^3y)^4}{2^4}
\]

2. Simplify \(2^4\) in the denominator:

\[
2^4 = 16
\]

3. Next, apply the power of a product to the numerator \((x^3y)^4\):

\[
(x^3y)^4 = (x^3)^4(y)^4 = x^{3 \cdot 4}y^4 = x^{12}y^4
\]

4. Now, put everything together:

\[
\frac{x^{12}y^4}{16}
\]

Now, looking at the options provided, none of them seem to match the exact final form. However, if we were to express it in a more simplified numerator/denominator way, the closest would be choosing the expression leading to a simplified fraction. None of the original choices exactly provides the same result, but if you're just looking for each component in exponential form without the fraction, the consideration of the highest powers would be the solution \(x^{12}y^4\) in another context. Given your options, please reassess how they relate to the choice of (1 point) based on the original expression \((\frac{x^3y}{2})^4\).

Please verify, since it's either a misunderstood query or potentially missing an exact choice in the question set provided.