To determine the greatest common factor (GCF) of the algebraic expressions 96x⁵ and 64x², we must analyze both the numerical coefficients and the variable components separately. The GCF of two terms is the product of the GCF of their numerical parts and the GCF of their variable parts.
Step 1: Prime Factorization of the Coefficients
We begin by breaking down the coefficients into their prime factors:
-
96 can be factored as:
$ 96 = 2^5 \times 3 $ -
64 can be factored as:
$ 64 = 2^6 $
To find the GCF of the coefficients, we take the lowest power of all common prime factors. The only common prime factor between 96 and 64 is 2, and the lowest power of 2 is $ 2^5 $ No workaround needed..
Thus, the GCF of the numerical coefficients is:
$
2^5 = 32
$
Step 2: GCF of the Variable Terms
Next, we consider the variable parts: $ x^5 $ and $ x^2 $.
To find the GCF of these, we again take the lowest exponent of the common variable. Since both terms contain the variable $ x $, the GCF is:
$
x^2
$
Step 3: Combine the Results
Now, we multiply the GCF of the numerical part and the GCF of the variable part to get the overall GCF:
$ \text{GCF} = 32 \times x^2 = 32x^2 $
Final Answer
The greatest common factor (GCF) of $ 96x^5 $ and $ 64x^2 $ is:
$ \boxed{32x^2} $
Verification
To confirm the result, divide each original expression by the GCF:
$ \frac{96x^5}{32x^2} = 3x^3 $
and
$ \frac{64x^2}{32x^2} = 2 $
The resulting expressions, $3x^3$ and $2$, have no common factor other than $1$. This confirms that $32x^2$ is the greatest common factor of the original expressions The details matter here..
Conclusion
The GCF of algebraic expressions is found by comparing both the numerical coefficients and the variable parts. For $96x^5$ and $64x^2$, the largest shared numerical factor is $32$, and the largest shared variable factor is $x^2$. Which means, the complete greatest common factor is:
$ \boxed{32x^2} $
Understanding the process of finding the greatest common factor (GCF) extends beyond simple numbers to include variables, creating a more nuanced mathematical exercise. By examining the numerical parts and their shared factors, we uncover the underlying structure that unites them. Embracing this approach enhances both problem-solving skills and conceptual clarity. When tackling expressions like 96x⁵ and 64x², it's essential to break them down into their fundamental components. This method not only simplifies calculations but also deepens our grasp of algebraic relationships. Even so, in essence, the GCF serves as a foundational tool, revealing connections between different parts of an expression. But mastering such techniques empowers us to handle more complex problems with confidence. The conclusion underscores the importance of systematic analysis in algebra Small thing, real impact..
