Question

question
which statement about the rectangular prism’s volume is correct?

• its volume must be expressed as the product of a whole number and a square root because its side lengths include both.

• in its simplest form, the volume can be expressed as the square root of a whole number that is not a perfect square.

• its volume can be expressed as a whole number because the product of the radicands is a perfect square.

• in its simplest form, the volume must be expressed as the product of two whole numbers.

Explanation:

To solve this, we analyze the volume of a rectangular prism (\(V = l \times w \times h\)) and the properties of square roots:

Analyzing Each Option:

1. First Option: The volume of a rectangular prism with side lengths including whole numbers and square roots is the product of those lengths, not necessarily a whole number times a square root in a way that defines "must." This is too restrictive. Eliminate.
2. Second Option: For the volume to be a square root of a non - perfect - square whole number, the product of the side lengths (with square roots) would need to simplify to \(\sqrt{n}\) (\(n\) non - perfect square). But if side lengths include whole numbers and square roots, the product is more likely a whole number (if radicands multiply to a perfect square) or a whole number times a square root, not just \(\sqrt{n}\). Eliminate.
3. Third Option: If the side lengths have square roots (e.g., \(\sqrt{a}\), \(\sqrt{b}\), \(\sqrt{c}\)) and whole numbers, the volume is \(l\times w\times h\). If the product of the radicands (\(a\times b\times c\)) is a perfect square, then \(\sqrt{a}\times\sqrt{b}\times\sqrt{c}=\sqrt{abc}\), and if \(abc\) is a perfect square, \(\sqrt{abc}\) is a whole number. So when multiplied by other whole - number side lengths (if any), the total volume can be a whole number. This matches the logic: if radicands multiply to a perfect square, square roots simplify to whole numbers, and the product of whole numbers is a whole number.
4. Fourth Option: The volume being the product of two whole numbers is not a general rule. A rectangular prism can have three side lengths (e.g., \(2\), \(3\), \(\sqrt{4}\) – here volume is \(2\times3\times2 = 12\), a product of three whole numbers in effect, not two). Eliminate.

Answer:

Its volume can be expressed as a whole number because the product of the radicands is a perfect square. (The option with this statement)