Identify the greatest common factor of [tex]$24 b w^3 z^2$[/tex] and [tex]$36 b^4 w^2 x$[/tex].
Answer (2)
Find the greatest common factor (GCF) of the coefficients 24 and 36, which is 12.
Identify the lowest power of the common variable b , which is b 1 = b .
Identify the lowest power of the common variable w , which is w 2 .
Multiply the GCF of the coefficients and the variables: 12 b w 2 .
Explanation
Problem Analysis We are asked to find the greatest common factor (GCF) of two terms: 24 b w 3 z 2 and 36 b 4 w 2 x . The GCF is the largest expression that divides both terms without leaving a remainder.
GCF of Coefficients First, we find the GCF of the coefficients, 24 and 36. The result of this operation is 12.
GCF of Variable b Next, we find the GCF of the variable b . The first term has b 1 and the second term has b 4 . The GCF is b 1 (or simply b ), since it is the lowest power of b present in both terms.
GCF of Variable w Now, we find the GCF of the variable w . The first term has w 3 and the second term has w 2 . The GCF is w 2 , since it is the lowest power of w present in both terms.
Examine Variables z and x The variable z only appears in the first term, and the variable x only appears in the second term. Therefore, they are not part of the GCF.
Combine the GCFs Finally, we multiply the GCF of the coefficients and the variables to find the overall GCF: 12 ⋅ b ⋅ w 2 = 12 b w 2 .
Final Answer Therefore, the greatest common factor of 24 b w 3 z 2 and 36 b 4 w 2 x is 12 b w 2 .
Examples
Understanding the greatest common factor is useful in simplifying algebraic expressions and in various real-world scenarios. For instance, imagine you are tiling a rectangular floor. You have tiles of size 24 b w 3 z 2 and 36 b 4 w 2 x , where b and w represent the dimensions of the tiles and x and z represent other tile characteristics. Finding the GCF, 12 b w 2 , helps you determine the largest tile size that can evenly divide both types of tiles, ensuring you can efficiently use the tiles without needing to cut them into smaller, uneven pieces. This simplifies the tiling process and reduces waste.
The greatest common factor of 24 b w 3 z 2 and 36 b 4 w 2 x is 12 b w 2 . We found this by determining the GCF of the coefficients and the lowest powers of the common variables. The final result combines these elements to give the GCF as 12 b w 2 .
;
