If the three algebraic terms 9b^2, 3a^2 and 4ab are multiplied together, what is the resulting simplified product in terms of a and b?

Introduction / Context:
Algebraic expressions involving products of powers are a core part of aptitude and school level mathematics. Questions like this test your comfort with multiplying coefficients and adding exponents with the same base. Here, three terms in a and b are multiplied, and you must simplify the product correctly using the rules of indices. This is a straightforward but important skill for both higher algebra and many competitive exams.

Given Data / Assumptions:

• The three terms are 9b^2, 3a^2 and 4ab.
• We assume a and b are algebraic variables.
• We need to find the simplified product 9b^2 * 3a^2 * 4ab.
• We will use the rule that a^m * a^n = a^(m+n).

Concept / Approach:
To multiply algebraic terms, first multiply all numerical coefficients, then combine like bases by adding their exponents. This problem has variables a and b. We will count the powers of a and b separately. Careful organisation of factors helps avoid missing any exponent, especially when more than two terms are involved. The goal is to arrive at a single monomial of the form constant times a to some power times b to some power.

Step-by-Step Solution:
Step 1: Write the full product explicitly: 9b^2 * 3a^2 * 4ab. Step 2: Multiply the numerical coefficients: 9 * 3 * 4 = 108. Step 3: Collect all a terms: there is a^2 from 3a^2 and a from 4ab, so together that is a^(2+1) = a^3. Step 4: Collect all b terms: there is b^2 from 9b^2 and b from 4ab, so together that is b^(2+1) = b^3. Step 5: Combine the results to get the simplified product: 108a^3b^3.

Verification / Alternative check:
As a quick check, you can rearrange the product grouping all a factors and b factors: (9 * 3 * 4) * (a^2 * a) * (b^2 * b). This gives 108 * a^3 * b^3, which matches the earlier calculation. Another way is to substitute simple values like a = 1 and b = 1. The original product becomes 9 * 3 * 4 = 108, and the simplified expression 108a^3b^3 also becomes 108, confirming the algebra is correct for this test case.

Why Other Options Are Wrong:

• Option a (108a^2b^2) ignores the extra powers of a and b contributed by 4ab.
• Option c (108a^3b^2) adds the exponents of a correctly but fails to add both b powers.
• Option d (108a^4b^4) overcounts the exponents, as there are only three factors of a and three of b, not four each.
• Option e (36a^3b^3) miscalculates the numerical coefficient, using 9 * 3 * 4 divided by 3 instead of the full product 108.

Common Pitfalls:
A common mistake is to multiply exponents instead of adding them, or to forget one of the variable factors, especially when both a and b occur in different terms. Another frequent error is to mishandle the numerical coefficient and perform partial multiplication. To avoid such mistakes, it is helpful to write out all parts of the product, group variables with the same base, and then add exponents carefully. Practising these steps builds accuracy and speed in algebraic manipulation.

Final Answer:
The simplified product of 9b^2, 3a^2 and 4ab is 108a^3b^3.