Question

1. a rectangular prism has length ( x ), width ( \frac{x}{2} ), and height ( \frac{x}{3} ). which of the expressions represent the 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:

Step1: Recall the volume formula for a rectangular prism

The volume \( V \) of a rectangular prism is given by the product of its length, width, and height. So, \( V=\text{length}\times\text{width}\times\text{height} \).

Step2: Substitute the given values

Given length \( = x \), width \(=\frac{x}{2}\), height \(=\frac{x}{3}\). Then,
\[

$$\begin{align*} V&=x\times\frac{x}{2}\times\frac{x}{3}\\ &=\frac{x\times x\times x}{2\times 3}\\ &=\frac{x^{3}}{6} \end{align*}$$

\]

Step3: Rewrite \(\frac{x^{3}}{6}\) using negative exponents

We know that \( \frac{1}{a}=a^{-1} \) and \( \frac{1}{ab}=a^{-1}b^{-1} \) (where \( a,b
eq0 \)).

• For option A: \( 6^{-1}x^{3}=\frac{1}{6}x^{3}=\frac{x^{3}}{6} \), which matches our volume formula.
• For option B: \( 6^{-1}x^{-1}=\frac{1}{6x} \), which is not equal to \( \frac{x^{3}}{6} \), so B is incorrect.
• For option C: First, simplify \( (6x^{-3})^{-1} \). Using the power - of - a - product rule \( (ab)^{n}=a^{n}b^{n} \) and the power - of - a - power rule \( (a^{m})^{n}=a^{mn} \), we have \( (6x^{-3})^{-1}=6^{-1}(x^{-3})^{-1}=6^{-1}x^{3}=\frac{x^{3}}{6} \), which matches our volume formula.
• For option D: \( 2^{-1}\cdot3^{-1}\cdot x^{3}=\frac{1}{2}\times\frac{1}{3}\times x^{3}=\frac{x^{3}}{6} \), which matches our volume formula.

Answer:

A. \(6^{-1}x^{3}\), C. \((6x^{-3})^{-1}\), D. \(2^{-1}\cdot3^{-1}\cdot x^{3}\)