1. The problem asks to represent the phrase "The cube of the product of 8 and a number" as an algebraic expression. 2. Let the number be represented by the variable $x$. 3. The product of 8 and the number $x$ is written as $8x$. 4. The cube of a quantity means raising it to the power of 3, so the cube of the product $8x$ is written as $(8x)^3$. 5. Therefore, the algebraic expression for the phrase is: $$ (8x)^3 $$ This expression is not simplified as per the problem instructions.