Problem 31 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of terms. $$ 18 a^{4}, 9 a^{3}, 27 a^{3} $$

Step-by-Step Solution

Answer

The GCF of the terms is \(9a^3\).

Step 1: Identify the Numerical GCD

First, break down each coefficient into its prime factors. For the given terms:- **18** can be factored as: \(2 imes 3^2\)- **9** can be factored as: \(3^2\)- **27** can be factored as: \(3^3\)All terms have the prime number \(3\), appearing at least in the power of 2. Therefore, the numerical greatest common divisor (GCD) is \(3^2 = 9\).

Step 2: Identify the Common Variable Factors

Next, explore the powers of \(a\) in each term:- For \(18a^4\), the power of \(a\) is \(a^4\).- For \(9a^3\), the power of \(a\) is \(a^3\).- For \(27a^3\), the power of \(a\) is \(a^3\).The smallest power of \(a\) that appears in all terms is \(a^3\). Thus, the GCD of the variables is \(a^3\).

Step 3: Combine the Numerical and Variable GCDs

Combine the numerical GCD with the variable GCD to determine the overall greatest common factor:GCD of coefficients: \(9\)GCD of variables: \(a^3\)Thus, the overall GCF is \(9a^3\).

Key Concepts

Understanding Numerical GCD

The concept of the Numerical Greatest Common Divisor (GCD) involves finding the highest common factor shared by numerical coefficients. Let's break this down into simpler terms.
The numerical GCD is essentially the "largest number" that divides each of the given numbers completely without leaving a remainder. In this particular exercise:

• 18 can be rewritten as a product of its prime numbers: \(18 = 2 \times 3^2\)
• 9 can be rewritten as \(9 = 3^2\)
• 27 can be rewritten as \(27 = 3^3\)

All these numbers share the common prime factor "3". To find the numerical GCD, identify the smallest power of common prime factors shared by all terms. Here, each term has at least two 3's, or \(3^2\), which gives us a numerical GCD of 9. It's like finding the largest slice size that can evenly slice all your pieces of cake!

Remember this method anytime you need to find a numerical GCD: break down each number into prime factors, and pick the smallest power common to all.

Exploring Variable GCD

When dealing with algebraic terms, unlike just numbers, finding out the GCD involves considering not only numbers but also variables. Here, we're looking at terms with the variable "a", each raised to different powers.
To find the variable GCD, you simply need to check the powers of the variable in each term. It's very straightforward:

• In \(18a^4\), the power of "a" is 4.
• In \(9a^3\), the power of "a" is 3.
• In \(27a^3\), the power of "a" is also 3.

The smallest power of the variable "a" that occurs in all terms is the key. Here, it's \(a^3\). Thus, the variable GCD is \(a^3\). This means that each term has at least three "a's" to share. When combining this logic with numerical GCD, it helps in problems where both coefficients and variables matter.

Prime Factorization: The Backbone of GCD

Prime factorization is like looking at a number under a powerful magnifying glass to see its basic building blocks, the prime numbers. In any GCD problem, breaking down numbers into their prime factors is crucial.
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. In our GCD problem above, we tackled prime factorization to strip down each coefficient into its simplest form:

• 18: Factored into \(2 \times 3^2\)
• 9: Factored into \(3^2\)
• 27: Factored into \(3^3\)

Breaking down these numbers by identifying 2 and 3 as prime numbers helps us pinpoint the smallest common power shared by these primes, which eventually gives us our numerical GCD.
This "backbone" allows us to understand not just individual numbers, but also how they relate to one another, and how we can calculate factors shared across different terms. Prime factorization is a key skill that helps to tackle both simple numerical problems and more complex algebraic equations alike.