find the greatest common factor and simplify your answer. 4v to the 7th power y to the 9th power and 16v to the 6th power x to the 8th power y to the 5th power.
To find the greatest common factor (GCF), we need to factorize both terms and find the common factors.
Let's factorize the first term, 4v^7y^9:
4v^7y^9 = 2^2 * v^7 * y^9
Now, let's factorize the second term, 16v^6x^8y^5:
16v^6x^8y^5 = 2^4 * v^6 * x^8 * y^5
To find the GCF, we take the smallest exponent for each common factor:
GCF = 2^2 * v^6 * y^5 = 4v^6y^5
So, the greatest common factor is 4v^6y^5.
To find the greatest common factor (GCF) and simplify the expression, we need to break down the given terms into their prime factors.
Let's start by factorizing each term individually:
1. Factorizing 4v^7y^9:
- 4 can be factored as 2 * 2.
- For v^7, there are no factors other than v repeated 7 times.
- For y^9, there are no factors other than y repeated 9 times.
Therefore, 4v^7y^9 = (2 * 2) * v * v * v * v * v * v * v * y * y * y * y * y * y * y * y * y.
2. Factorizing 16v^6x^8y^5:
- 16 can be factored as 2 * 2 * 2 * 2.
- For v^6, there are no factors other than v repeated 6 times.
- For x^8, there are no factors other than x repeated 8 times.
- For y^5, there are no factors other than y repeated 5 times.
Therefore, 16v^6x^8y^5 = (2 * 2 * 2 * 2) * v * v * v * v * v * v * x * x * x * x * x * x * x * x * y * y * y * y * y.
Now, let's find the common factors among these terms.
The common factors are:
- 2 (appeared four times in each term)
- v (appeared six times in both terms)
- y (appeared five times in both terms)
To find the greatest common factor (GCF), we multiply these common factors together:
GCF = 2 * 2 * 2 * 2 * v * v * v * v * v * v * y * y * y * y * y
Simplifying this expression:
GCF = 16v^6y^5
Therefore, the greatest common factor is 16v^6y^5.
GCF of
4v7y9 and
16v6x8y5
Take the lower power of the same base:
GCF=4v6y5