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The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.?
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The cube of the difference between two given natural numbers is 1728, ...
Step 1: Understanding the Problem
To solve the problem, we have two natural numbers, which we can denote as x and y. The conditions given are:
- The cube of the difference between the two numbers: (x - y)³ = 1728
- The product of the two numbers: x * y = 108
Step 2: Finding the Difference
First, we solve for the difference:
- From (x - y)³ = 1728, we find that x - y = ∛1728 = 12.
Step 3: Setting Up Equations
Next, we have:
- x - y = 12 (1)
- x * y = 108 (2)
We can express x in terms of y using equation (1):
- x = y + 12
Now, substitute this expression into equation (2):
- (y + 12) * y = 108
Step 4: Solving for y
Expanding the equation:
- y² + 12y - 108 = 0
Now we can factor or use the quadratic formula. Factoring gives us:
- (y + 18)(y - 6) = 0
Thus, y = 6 (since y must be a natural number).
Step 5: Finding x
Now substitute y back to find x:
- x = y + 12 = 6 + 12 = 18.
Step 6: Finding the Sum of the Cubes
Now, we need to find the sum of the cubes of x and y:
- x³ + y³ = 18³ + 6³ = 5832 + 216 = 6048.
Conclusion
The sum of the cubes of the two given natural numbers is 6048.
To solve the problem, we have two natural numbers, which we can denote as x and y. The conditions given are:
- The cube of the difference between the two numbers: (x - y)³ = 1728
- The product of the two numbers: x * y = 108
Step 2: Finding the Difference
First, we solve for the difference:
- From (x - y)³ = 1728, we find that x - y = ∛1728 = 12.
Step 3: Setting Up Equations
Next, we have:
- x - y = 12 (1)
- x * y = 108 (2)
We can express x in terms of y using equation (1):
- x = y + 12
Now, substitute this expression into equation (2):
- (y + 12) * y = 108
Step 4: Solving for y
Expanding the equation:
- y² + 12y - 108 = 0
Now we can factor or use the quadratic formula. Factoring gives us:
- (y + 18)(y - 6) = 0
Thus, y = 6 (since y must be a natural number).
Step 5: Finding x
Now substitute y back to find x:
- x = y + 12 = 6 + 12 = 18.
Step 6: Finding the Sum of the Cubes
Now, we need to find the sum of the cubes of x and y:
- x³ + y³ = 18³ + 6³ = 5832 + 216 = 6048.
Conclusion
The sum of the cubes of the two given natural numbers is 6048.
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The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.?
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The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.? for SSC CHSL 2025 is part of SSC CHSL preparation. The Question and answers have been prepared according to the SSC CHSL exam syllabus. Information about The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.? covers all topics & solutions for SSC CHSL 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.?.
The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.? for SSC CHSL 2025 is part of SSC CHSL preparation. The Question and answers have been prepared according to the SSC CHSL exam syllabus. Information about The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.? covers all topics & solutions for SSC CHSL 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.?.
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