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Understanding how to divide numerical coefficients is a crucial skill when simplifying algebraic expressions. When you come across a division problem involving monomials, the first step is to divide the numbers in front of the variables, these are called the numerical coefficients. To divide numerical coefficients, simply treat them as you would with any other basic division problem.

For example, if you have \( -15x^{16}y^{2} \) divided by \( 45x^{2}y^{2} \) like in our exercise, look only at the numerical coefficients, \( -15 \) and \( 45 \) initially. When we divide \( -15 \) by \( 45 \) we get \( -\frac{15}{45} \), which simplifies to \( -\frac{1}{3} \). Simplification is possible here because 15 and 45 have a common factor; in this case, it's 15. Therefore, dividing both by 15, we find that \( -15 \) becomes \( -1 \) and \( 45 \) becomes \( 3 \) resulting in the simplified coefficient for our quotient.

The division of variables in algebra follows specific exponent rules. When dividing variables with the same base, you subtract the exponent (or power) of the divisor from the exponent of the dividend. This exponent subtraction is based on the 'quotient of powers' rule, which states that when you divide powers with the same base, you can keep the base and subtract exponents.

Using our example, we divide \( x^{16} \) by \( x^{2} \) by subtracting their exponents. Mathematically, this is represented as \( x^{16-2} = x^{14} \). It is important to note that this rule only applies to variables with the same base; in this case, the base is \( x \). Similarly, when the exponents are equal, such as \( y^{2} \) divided by \( y^{2} \) the result is \( y^{2-2} = y^{0} \), which equals 1, since any nonzero number raised to the power of 0 is 1.

To verify the correctness of a division of monomials, it's a good practice to check the answer by multiplication. This means multiplying the divisor by the quotient to see if you get the original dividend. Good checking helps to confirm that no mistakes were made during the division process.

In our exercise, checking involves taking our divisor, \( 45x^{2}y^{2} \), and multiplying it by the quotient, \( -\frac{1}{3}x^{14} \). When we perform the multiplication, we should get \( -15x^{16}y^{2} \) which would match our original dividend. If the result is the same, then you can be confident that your initial division was done correctly.

This step is essential not only to check your work but also to reinforce your understanding of monomial operations and their properties. Remember, the order of operations should still be followed when checking; multiply coefficients first and then apply the 'product of powers' rule for variables with the same base.