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Find the positive integer n such that the expansion of (4x^2 - 7y^3)^n contains a term of the form cx^2*y^6.

 Jun 19, 2023
 #1
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I suggest you go to this link. : https://web2.0calc.com/questions/counting_38343

 Jun 19, 2023
 #2
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Anyways,      \(\boxed{n=3}\)

 Jun 19, 2023
 #3
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The question isn't the same tho in that link. 

 Jun 19, 2023
 #4
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The term cx^2y^6 will be present in the expansion of (4x^2 - 7y^3)^n if and only if n is divisible by 2 and 3. This is because the term cx^2y^6 can only be created by combining two factors of x^2 and three factors of y^3.

The smallest positive integer n that satisfies these conditions is n = 6. This is because 6 is divisible by 2 and 3, and the expansion of (4x^2 - 7y^3)^6 contains the term 4x^2y^6.

Here is a more detailed explanation of why n must be divisible by 2 and 3.

The term cx^2*y^6 can only be created by combining two factors of x^2 and three factors of y^3. This means that there must be at least two instances of x^2 and at least three instances of y^3 in the expansion of (4x^2 - 7y^3)^n.

The expansion of (4x^2 - 7y^3)^n contains a term of x^2 in the first term, which is 4x^2. This term will be present in the expansion regardless of the value of n.

The expansion of (4x^2 - 7y^3)^n contains a term of y^3 in the third term, which is -7y^3. This term will be present in the expansion regardless of the value of n.

For the term cx^2*y^6 to be present in the expansion, there must be at least one other term in the expansion that contains a factor of x^2 and a factor of y^3.

The only way for this to happen is if n is divisible by 2 and 3. If n is not divisible by 2, then there will be no other terms in the expansion that contain a factor of x^2. If n is not divisible by 3, then there will be no other terms in the expansion that contain a factor of y^3.

Therefore, n must be divisible by 2 and 3 in order for the term cx^2*y^6 to be present in the expansion.

 Jun 19, 2023
 #5
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No, it isn't. n=6. Expand: (4x^2 - 7y^3)^6 ==4096 x^12 - 43008 x^10 y^3 + 188160 x^8 y^6 - 439040 x^6 y^9 + 576240 x^4 y^12 - 403368 x^2 y^15 + 117649 y^18 - where is:x^2 y^6

 

n==3.  Expand:  (4x^2 - 7y^3)^3 ==64 x^6 - 336 x^4 y^3 + 588 x^2 y^6 - 343 y^9

Guest Jun 19, 2023
 #7
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Why should we help you? Why are you more important than our own problems? Sorry for bursting your little bubble, but you aren't the center of the universe. Now shut it and wait for your turn.

 Jun 20, 2023

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