- Calculate the existing GCF of the given terms: \(\gcd(36h^3, 12h^6) = 12h^3\).
- Recognize that for a new term \(X\), the overall GCF remains \(12h^3\) if and only if \(\gcd(12h^3, X) = 12h^3\), meaning \(12h^3\) must be a factor of \(X\).
- Test the options: \(6h^3, 12h^2, 30h^4\) are not divisible by \(12h^3\), whereas \(48h^5 = 12h^3 \times 4h^2\).
- Select the correct term: \(\boxed{48h^5}\).
Explanation
- Problem Analysis and GCF Requirement
The objective is to find which of the options (\(6h^3, 12h^2, 30h^4, 48h^5\)) added to the list \(\{36h^3, 12h^6\}\) results in a Greatest Common Factor (GCF) of \(12h^3\). The GCF of a set of terms is found by identifying the greatest common divisor of the numerical coefficients and the lowest power of each common variable present in all terms. Given the terms \(36h^3\) and \(12h^6\), we first check their GCF: the coefficients \(36\) and \(12\) have a GCF of \(12\), and the powers of \(h\) are \(h^3\) and \(h^6\), for which the lowest power is \(h^3\). Thus, \(\gcd(36h^3, 12h^6) = 12h^3\). We need to find a third term \(X\) such that \(\gcd(12h^3, 12h^6, X) = 12h^3\). Since \(\gcd(36h^3, 12h^6) = 12h^3\), this simplifies to finding \(X\) such that \(\gcd(12h^3, X) = 12h^3\). This condition is satisfied if and only if \(12h^3\) is a factor of \(X\).
- Evaluating the Options
We now test each of the provided options to see if \(12h^3\) is a factor of each candidate term \(X\):
- For \(X = 6h^3\): Is \(12h^3\) a factor of \(6h^3\)? No, because \(12\) does not divide \(6\).
- For \(X = 12h^2\): Is \(12h^3\) a factor of \(12h^2\)? No, because \(h^3\) does not divide \(h^2\).
- For \(X = 30h^4\): Is \(12h^3\) a factor of \(30h^4\)? No, because \(12\) does not divide \(30\).
- For \(X = 48h^5\): Is \(12h^3\) a factor of \(48h^5\)? Yes, because \(48 = 12 \times 4\) and \(h^5 = h^3 \times h^2\). Specifically, \(48h^5 = (12h^3) \times (4h^2)\). Since \(12h^3\) divides \(48h^5\), it follows that \(\gcd(12h^3, 48h^5) = 12h^3\).
- Final Conclusion and Selection
Comparing the results, the only term that satisfies the condition \(\gcd(12h^3, 12h^6, X) = 12h^3\) is \(48h^5\). Therefore, the correct term to add to the list is \(48h^5\).
Examples
In industrial manufacturing, parts are often grouped into batches that share specific design features. Suppose you have two batches of components: one batch is labeled with a feature power of \(36h^3\) and the other with \(12h^6\). To efficiently machine these parts, you want to find a third batch that keeps the common tool adjustment parameter, defined as the greatest common factor (GCF) of the batch identifiers, at exactly \(12h^3\). By identifying that \(48h^5\) is the correct addition, you ensure that the universal manufacturing tool settings remain consistent across all three batches of parts.