Find the greatest common factor (highest common factor) of the algebraic terms 24b⁶c⁸d², 18a⁶c²d⁴, and 12a⁴b⁴ by comparing coefficients and the powers of each variable.

Introduction / Context:
This algebra question tests understanding of how to find the highest common factor (HCF) of algebraic expressions. The HCF is the largest expression that divides each term exactly. Here we must examine numeric coefficients and variable powers in three given terms. Such skills are important in factorisation and simplification tasks in algebra.

Given Data / Assumptions:

• The three terms are 24b⁶c⁸d², 18a⁶c²d⁴, and 12a⁴b⁴.
• We must find the highest common factor that divides each term.
• The HCF is built from the greatest common divisor of the coefficients and the lowest power of each variable that appears in all three terms.

Concept / Approach:
To find the HCF of algebraic terms, follow two steps. First, compute the greatest common divisor of the numerical coefficients. Second, for each variable, identify its smallest exponent among all terms where that variable appears in every term. If a variable is missing from any term, it does not appear in the HCF. Multiplying the numerical part and the chosen variable factors gives the required HCF.

Step-by-Step Solution:
Look at the coefficients: 24, 18, and 12.The greatest common divisor of 24, 18, and 12 is 6.Now examine variable a: powers are 0 in the first term, 6 in the second, and 4 in the third.Since a has power 0 in the first term, a does not appear in every term, so a is not part of the HCF.Next variable b: powers are 6, 0, and 4.Again, b is missing (power 0) in the second term, so b is not part of the HCF.For c: powers are 8, 2, and 0.c is missing from the third term, so c is not in the HCF.For d: powers are 2, 4, and 0.d is missing from the third term, so d is not included in the HCF.Therefore, the HCF is simply the numeric factor 6.

Verification / Alternative check:
Check divisibility. Each coefficient 24, 18, and 12 is divisible by 6. No larger integer greater than 6 divides all three coefficients, so 6 is the greatest numerical factor. Since no variable is common to all three terms, any variable factor would fail to divide at least one term. Thus 6 is indeed the largest common factor of all three expressions.

Why Other Options Are Wrong:
Options involving variables such as 72a²b²c²d², 72a⁶b⁶c⁸d⁴, 6a²b², and 12ab² all contain variables a, b, c, or d. Each of these variables is missing in at least one of the terms, so these expressions cannot divide every term exactly. Furthermore, coefficients like 72 or 12 are not the greatest common divisor of 24, 18, and 12. Therefore those options cannot represent the HCF.

Common Pitfalls:
A common mistake is to think that any variable appearing in two of the terms must be included in the HCF. However, the HCF requires that every factor must divide each term. Another error is to take the highest exponent of a variable instead of the lowest, which is actually used for the least common multiple, not the HCF. Always remember that for the HCF, numeric factors use greatest common divisor while variable factors use the smallest exponent across all terms that contain that variable.

Final Answer:
The highest common factor of the given algebraic terms is 6.