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The operation "!' is defined for the real nonzero numbers X and Y. It is known that X ! ( Y ! Z) = ( X ! Y) x Z and that X ! X = 1. What is the value of A if (15! A) = 3?
Correct answer is '5'. Can you explain this answer?
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The operation !is defined for the real nonzero numbers X and Y. It is ...
Answer: 5
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The operation !is defined for the real nonzero numbers X and Y. It is ...
Given:
The operation ! is defined for the real nonzero numbers X and Y. It is known that X ! ( Y ! Z) = ( X ! Y) x Z and that X ! X = 1.
To Find:
The value of A if (15! A) = 3.
Solution:
Let's break down the given information and use it to find the value of A.
1. X ! (Y ! Z) = (X ! Y) x Z
This equation tells us how the operation ! behaves when applied to three numbers X, Y, and Z. The operation is associative, meaning that it doesn't matter how we group the numbers. We can first apply the operation to Y and Z, and then apply it to X and the result, or we can first apply it to X and Y, and then apply it to the result and Z. Both ways should give us the same result.
2. X ! X = 1
This equation tells us that when we apply the operation ! to a number X twice, we get the value 1. Essentially, it acts as an identity element for the operation !.
Using the given equations:
Let's substitute 15 for X and A for Z in the equation X ! (Y ! Z) = (X ! Y) x Z, as we need to find the value of A.
15 ! (Y ! A) = (15 ! Y) x A
Since we don't know the value of Y, we can't simplify the equation further. However, we know that X ! X = 1. So, let's substitute 15 for X in this equation:
15 ! 15 = 1
Now, let's substitute 15 for X in the first equation and simplify it:
15 ! (Y ! A) = (15 ! Y) x A
1 ! (Y ! A) = (1 ! Y) x A
(Y ! A) = A
Now, let's substitute this result back into the first equation:
1 ! A = (1 ! Y) x A
1 = (1 ! Y) x A
Since the operation ! is defined for nonzero real numbers, the only possible value for (1 ! Y) is 1. Therefore, the equation simplifies to:
1 = A
Therefore, the value of A is 1.
The operation ! is defined for the real nonzero numbers X and Y. It is known that X ! ( Y ! Z) = ( X ! Y) x Z and that X ! X = 1.
To Find:
The value of A if (15! A) = 3.
Solution:
Let's break down the given information and use it to find the value of A.
1. X ! (Y ! Z) = (X ! Y) x Z
This equation tells us how the operation ! behaves when applied to three numbers X, Y, and Z. The operation is associative, meaning that it doesn't matter how we group the numbers. We can first apply the operation to Y and Z, and then apply it to X and the result, or we can first apply it to X and Y, and then apply it to the result and Z. Both ways should give us the same result.
2. X ! X = 1
This equation tells us that when we apply the operation ! to a number X twice, we get the value 1. Essentially, it acts as an identity element for the operation !.
Using the given equations:
Let's substitute 15 for X and A for Z in the equation X ! (Y ! Z) = (X ! Y) x Z, as we need to find the value of A.
15 ! (Y ! A) = (15 ! Y) x A
Since we don't know the value of Y, we can't simplify the equation further. However, we know that X ! X = 1. So, let's substitute 15 for X in this equation:
15 ! 15 = 1
Now, let's substitute 15 for X in the first equation and simplify it:
15 ! (Y ! A) = (15 ! Y) x A
1 ! (Y ! A) = (1 ! Y) x A
(Y ! A) = A
Now, let's substitute this result back into the first equation:
1 ! A = (1 ! Y) x A
1 = (1 ! Y) x A
Since the operation ! is defined for nonzero real numbers, the only possible value for (1 ! Y) is 1. Therefore, the equation simplifies to:
1 = A
Therefore, the value of A is 1.
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The operation !is defined for the real nonzero numbers X and Y. It is known that X ! ( Y ! Z) = ( X ! Y) x Zand that X ! X = 1. What is the value of A if (15! A) = 3?Correct answer is '5'. Can you explain this answer?
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