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When multiplying monomials, we start with the straightforward step of multiplying coefficients. Coefficients are simply the numerical part of algebraic terms and when multiplied together give us the new coefficient for our resulting expression. For instance, taking the coefficients (-5) and (3) from the monomials (-5y^4) and (3y^3), we multiply them just like regular numbers. This simply means (-5) × (3) = -15. A key point to always remember is the sign rule: when you multiply two numbers with different signs, like in this case, the product will always have a negative sign. This step reduces the complexity of our algebraic problem down to a basic arithmetic operation, setting the stage for us to deal with the variable part of the monomials next.

Once we've dealt with the coefficients, it's time to handle the variables. Here is where the rules of exponents come into play. Exponents tell us how many times a number (or variable) is multiplied by itself. One core rule is that when multiplying terms with the same base, you keep the base and add the exponents. In the given example, we combine the exponents from the variables:
(y^4) and (y^3). According to the rules, we'd have y raised to the 4+3, which simplifies to y^7.

It's this fundamental concept that provides the logical step to compress repeated multiplication of the same factor into a more concise expression. In practice, this rule of exponents helps students simplify algebraic expressions efficiently, allowing them to tackle even more complex algebra problems.

Algebraic expressions are combinations of numbers, variables, and mathematical operators. Our focus when multiplying monomials is to transform the given expressions into their simplest form. The steps we've covered earlier — multiplying the coefficients and applying the rules of exponents — are both integral strategies for simplifying these expressions. In completing the exercise
(-5y^4)(3y^3), the multiplication of coefficients and addition of exponents result in the final simplified algebraic expression, (-15y^7).

This offers a prime example of how algebraic expressions, which may initially appear complex, are broken down through rules and operations into more manageable pieces. Mastering these steps fully can significantly improve a student's ability to work with and understand algebra at a deeper level.