Apply the formulas for volume of a sphere and a cube.

To understand the volume of a sphere and a cube, as well as to identify perfect cubes, let's break this down step by step.

1. Volume Formulas

• Volume of a Cube: The formula for the volume \( V \) of a cube with side length \( s \) is given by: \[ V = s^3 \]
• Volume of a Sphere: The formula for the volume \( V \) of a sphere with radius \( r \) is given by: \[ V = \frac{4}{3} \pi r^3 \]

2. Identifying Perfect Cubes

A perfect cube is a number that can be expressed as \( n^3 \) where \( n \) is an integer. Examples of perfect cubes include:

• \( 1^3 = 1 \)
• \( 2^3 = 8 \)
• \( 3^3 = 27 \)
• \( 4^3 = 64 \)
• \( 5^3 = 125 \)
• \( 6^3 = 216 \)
• \( 7^3 = 343 \)
• \( 8^3 = 512 \)
• \( 9^3 = 729 \)
• \( 10^3 = 1000 \)

3. Understanding Side Length and Volume Measures

To find the side length of a cube given its volume, you can rearrange the volume formula: \[ s = \sqrt[3]{V} \] And for a sphere, if you know its volume and you want to find the radius, you can rearrange the volume formula: \[ r = \sqrt[3]{\frac{3V}{4\pi}} \]

Summary

• Volume of a Cube: \( V = s^3 \)
• Volume of a Sphere: \( V = \frac{4}{3} \pi r^3 \)
• Perfect Cubes: Numbers like 1, 8, 27 are examples.
• Side Length Calculation: For a cube, \( s = \sqrt[3]{V} \) and for a sphere, \( r = \sqrt[3]{\frac{3V}{4\pi}} \).

This structure allows you to understand the relationships between volumes, perfect cubes, and their properties.