The following is based on my private practice as a math tutor in Las Vegas since 2008, as well as teaching experience in the classroom. Addition and multiplication of monomials requires the novice algebra student to shift gears between adding and multiplying coefficients, as well as determining when addition of exponents is in order. This is demonstrated in the image above. In the first case (in black) the sum of monomials requires the coefficients, 7 and 3, to be added, but not the exponents, 2 and 2. In the second case (in blue), the product of monomials requirest the coefficients, 7 and 3, to be multiplied, and the exponents, 2 and 2, to be added. Since the second case has a combination of multiplication (of coefficents) and addition (of exponents), it's understandable why novice algebra students can get mixed up. Through the repetition of correctly solving enough of these operations over time, students develop the proficiency in knowing when to add or multiply coefficients, and when to add exponents.
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Mr. Ken Marciel

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