Final answer:
The phrase 'the product of three, and a number increased by five' translates to the algebraic expression 3(x + 5). This follows the commutative property of multiplication and reflects basic rules regarding operations with integers and their signs.
Step-by-step explanation:
To translate "the product of three, and a number increased by five" into an algebraic expression, you identify 'the product' as a multiplication operation, 'a number' as a variable (let's use x), and 'increased by five' as addition to that variable. Therefore, the expression is 3(x + 5).
The multiplication rules in algebra state that the order in which we multiply numbers does not change the product (commutative property of multiplication). This is similar to addition, where A + B = B + A. Moreover, when dealing with exponents, such as in expressions like 32 * 35, you would add the exponents if the bases are the same, resulting in 37, which is following the rule xp * xq = x(p+q).
Remember that when two numbers with the same sign are multiplied, the result has a positive sign, and when they have opposite signs, the result will be negative.