Question

volume of the prism? select all that apply.
a. $6^{-1}x^{3}$
b. $6^{-1}x^{-3}$
c. $(6x^{-3})^{-1}$
d. $2^{-1}\cdot 3^{-1}\cdot x^{3}$

Explanation:

To solve this, we analyze each option using exponent rules (\(a^{-n}=\frac{1}{a^n}\), \((ab)^n=a^n b^n\), \((a^m)^n=a^{mn}\)):

Analyze Option A: \(6^{-1}x^3\)

\(6^{-1}=\frac{1}{6}\), so \(6^{-1}x^3 = \frac{x^3}{6}\).

Analyze Option B: \(6^{-1}x^{-3}\)

\(6^{-1}=\frac{1}{6}\) and \(x^{-3}=\frac{1}{x^3}\), so \(6^{-1}x^{-3}=\frac{1}{6x^3}\).

Analyze Option C: \((6x^{-3})^{-1}\)

Using \((ab)^n=a^n b^n\) and \((a^m)^n=a^{mn}\):
\((6x^{-3})^{-1}=6^{-1}(x^{-3})^{-1}=6^{-1}x^{(-3)(-1)}=6^{-1}x^3=\frac{x^3}{6}\) (matches Option A).

Analyze Option D: \(2^{-1}\bullet 3^{-1}\bullet x^3\)

\(2^{-1}=\frac{1}{2}\), \(3^{-1}=\frac{1}{3}\), so \(2^{-1}\bullet 3^{-1}\bullet x^3=\frac{1}{2}\bullet\frac{1}{3}\bullet x^3=\frac{x^3}{6}\) (matches Option A and C).

Assuming the volume simplifies to \(\frac{x^3}{6}\) (from context, e.g., if the prism’s volume involves \(6\) and \(x^3\) with negative exponents resolved), the correct options are:

Answer:

A. \(6^{-1}x^3\)
C. \((6x^{-3})^{-1}\)
D. \(2^{-1}\bullet 3^{-1}\bullet x^3\)