Final answer:
The greatest common factor of the expressions 2v^8(w-2) and 12v(w-2)^7 is 2v(w-2)^7.
Step-by-step explanation:
To find the greatest common factor (GCF) of the expressions 2v8(w-2) and 12v(w-2)7, we need to look for the highest powers of common variables and constants that appear in both expressions.
First, consider the numerical coefficients: the GCF of 2 and 12 is 2. Next, we look at the variable v. Since the lowest power of v appearing in both expressions is 1 (from the second expression), v is a part of our GCF. Lastly, the expression (w-2) is common to both, but we take the lowest power, which is 7 since it appears in the second expression.
Combining these findings, the GCF of the two expressions is 2v(w-2)7.
To find the greatest common factor (GCF) of the expressions 2v⁸(w-2) and 12v(w-2)⁷, we need to factorize both expressions.
For the first expression, we can factor out 2v⁸ and write it as 2v⁸(w-2) = 2v⁸w - 4v⁸.
For the second expression, we can factor out 12v and write it as 12v(w-2)⁷ = 12vw⁷ - 24v⁷.
Now, we can see that both expressions have a common factor of v(w-2). Therefore, the GCF of the two expressions is v(w-2).