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If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x)y)³ + 8(log(y)z)³ +27(log(z)x)³?
Most Upvoted Answer
If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x...
Understanding the Given Equation
The equation provided is:
log(x)y + 2log(y)z + 3log(z)x = 0
This can be rewritten as:
A + 2B + 3C = 0
where:
- A = log(x)y
- B = log(y)z
- C = log(z)x
Setting Up the Values
To find the value of (log(x)y)³ + 8(log(y)z)³ + 27(log(z)x)³, we can introduce variables:
- Let A = log(x)y
- Let B = log(y)z
- Let C = log(z)x
Now, we need to express the original equation in terms of A, B, and C:
A + 2B + 3C = 0
Finding the Relationship
From A + 2B + 3C = 0, we can express one variable in terms of the others. For example:
A = -2B - 3C
Calculating the Required Expression
We want to evaluate:
A³ + 8B³ + 27C³
Using the identity for sums of cubes:
A³ + 8B³ + 27C³ = (A + 2B + 3C)(A² + 4B² + 9C² - 2AB - 6AC - 3BC)
Since A + 2B + 3C = 0, the entire expression simplifies to:
A³ + 8B³ + 27C³ = 0
Conclusion
Thus, the value of (log(x)y)³ + 8(log(y)z)³ + 27(log(z)x)³ is:
0
The equation provided is:
log(x)y + 2log(y)z + 3log(z)x = 0
This can be rewritten as:
A + 2B + 3C = 0
where:
- A = log(x)y
- B = log(y)z
- C = log(z)x
Setting Up the Values
To find the value of (log(x)y)³ + 8(log(y)z)³ + 27(log(z)x)³, we can introduce variables:
- Let A = log(x)y
- Let B = log(y)z
- Let C = log(z)x
Now, we need to express the original equation in terms of A, B, and C:
A + 2B + 3C = 0
Finding the Relationship
From A + 2B + 3C = 0, we can express one variable in terms of the others. For example:
A = -2B - 3C
Calculating the Required Expression
We want to evaluate:
A³ + 8B³ + 27C³
Using the identity for sums of cubes:
A³ + 8B³ + 27C³ = (A + 2B + 3C)(A² + 4B² + 9C² - 2AB - 6AC - 3BC)
Since A + 2B + 3C = 0, the entire expression simplifies to:
A³ + 8B³ + 27C³ = 0
Conclusion
Thus, the value of (log(x)y)³ + 8(log(y)z)³ + 27(log(z)x)³ is:
0
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If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x)y)³ + 8(log(y)z)³ +27(log(z)x)³?
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If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x)y)³ + 8(log(y)z)³ +27(log(z)x)³? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x)y)³ + 8(log(y)z)³ +27(log(z)x)³? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x)y)³ + 8(log(y)z)³ +27(log(z)x)³?.
If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x)y)³ + 8(log(y)z)³ +27(log(z)x)³? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x)y)³ + 8(log(y)z)³ +27(log(z)x)³? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If log(x)y + 2log(y)z + 3log(z)x = 0, then what is the value of (log(x)y)³ + 8(log(y)z)³ +27(log(z)x)³?.
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