What is the highest common factor (HCF) of the three algebraic expressions 12a^4 b^6, 18a^6 c^2, and 36a^2 b^2?

Introduction / Context:
This problem asks for the highest common factor (HCF) of three algebraic monomials. To find the HCF, you must identify the greatest common divisor of the numerical coefficients and the lowest power of each variable that is present in all terms. This is directly analogous to finding the HCF of integers, but extended to also consider variable exponents.

Given Data / Assumptions:

• The expressions are 12a^4 b^6, 18a^6 c^2, and 36a^2 b^2.
• We want the highest common factor shared by all three terms.
• All variables a, b, and c represent real numbers, but the HCF is expressed symbolically.

Concept / Approach:
To find the HCF of algebraic terms, we do two things. First, find the greatest common divisor of the numerical coefficients. Second, for each variable, take the smallest exponent that appears among the three terms, because a common factor cannot contain a higher power of a variable than any of the given terms. Variables that do not appear in all the expressions are not included in the HCF at all.

Step-by-Step Solution:
Consider the coefficients 12, 18, and 36.The HCF of 12, 18, and 36 is 6 because 6 divides each number and no larger integer does so.Now analyse the powers of a in the three terms: a^4, a^6, and a^2.The smallest exponent of a among 4, 6, and 2 is 2, so the HCF includes a^2.Next, look at b: in the three terms, we have b^6, b^0 (since b is absent in 18a^6 c^2), and b^2.The smallest exponent among 6, 0, and 2 is 0, meaning b does not appear in every term, so no b factor is in the HCF.Finally, examine c: the exponents are c^0, c^2, and c^0; the smallest exponent is again 0, so c is not part of the HCF.Therefore the HCF is 6a^2.

Verification / Alternative check:
We can verify this by dividing each term by 6a^2. For 12a^4 b^6, dividing by 6a^2 gives 2a^2 b^6. For 18a^6 c^2, dividing by 6a^2 gives 3a^4 c^2. For 36a^2 b^2, dividing by 6a^2 gives 6b^2. All quotients are still algebraic monomials with integer coefficients and non negative exponents, showing that 6a^2 divides each term. If we tried a larger factor such as 6a^2b^2, it would fail to divide 18a^6 c^2 because that term contains no factor of b at all.

Why Other Options Are Wrong:

• 36a^2 has a coefficient too large; 36 does not divide 18 evenly as part of the common factor.
• 108b^2 does not even contain a factor of a and, moreover, 108 does not divide 12 or 18.
• 6a^2b^2 incorrectly includes b^2, which is missing from 18a^6 c^2.
• 12a^2b^2 also includes b^2 and has a coefficient that does not divide all three terms.

Common Pitfalls:

• Taking the largest exponent instead of the smallest when forming the HCF of variable parts.
• Including variables that are not present in all terms.
• Looking for a common multiple rather than a common factor, which is the opposite objective.

Final Answer:
6a^2