Question

1. the volume of an ice cube is \\(32\\,000\text{ mm}^3\\). the exact length of each edge of the ice cube can be written in simplest mixed radical form as \\(p\sqrt3{q}\\) where \\(p\\) and \\(q\\) are whole numbers.

the value of \\(p - q\\) is ______.

(record your answer in the numerical response box from left to right.)

Explanation:

Set up the volume equation

The volume \(V\) of a cube with edge length \(s\) is given by:

$$V = s^3$$

We are given that the volume is \(32\,000\text{ mm}^3\). Thus, we set up the equation:

$$s^3 = 32\,000$$

Solve for the edge length

To find the edge length \(s\), we take the cube root of both sides:

$$s = \sqrt[3]{32\,000}$$

Simplify the radical expression

We factor \(32\,000\) to find perfect cube factors:

$$32\,000 = 8 \times 4 \times 1000$$

Since \(8 = 2^3\) and \(1000 = 10^3\), we can simplify the radical:

$$s = \sqrt[3]{8 \times 1000 \times 4}$$

$$s = \sqrt[3]{8} \times \sqrt[3]{1000} \times \sqrt[3]{4}$$

$$s = 2 \times 10 \times \sqrt[3]{4}$$

$$s = 20\sqrt[3]{4}$$

Identify the values of p and q

The simplified mixed radical form is given as \(p\sqrt[3]{q}\). Comparing this to our result:

$$p = 20$$

$$q = 4$$

Calculate the final value

We need to find the value of \(p - q\):

$$p - q = 20 - 4 = 16$$

Answer:

The volume of an ice cube is \(32\,000\text{ mm}^3\). The exact length of each edge of the ice cube can be written in simplest mixed radical form as \(p\sqrt[3]{q}\) where \(p\) and \(q\) are whole numbers.

The value of \(p - q\) is <blank>16</blank>.