The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the positive difference between the cubes of these two given numbers.
5616
Solving the Problem: Finding the Difference of Cubes
The problem asks us to find the positive difference between the cubes of two natural numbers. We are given two key pieces of information about these two numbers:
- The cube of their difference is 1728.
- Their product is 108.
Step-by-Step Solution
Step 1: Finding the Difference Between the Numbers
Let the two natural numbers be \(a\) and \(b\). We are given that the cube of their difference is 1728. This can be written mathematically as:
\[ (a-b)^3 = 1728 \]To find the difference \(a-b\), we need to take the cube root of both sides of the equation:
\[ a-b = \sqrt[3]{1728} \]We know that \(12 \times 12 \times 12 = 144 \times 12 = 1728\). Therefore, the cube root of 1728 is 12.
\[ a-b = 12 \]Since we are looking for the positive difference between the cubes, we assume \(a \gt b\), so \(a-b\) is positive.
Step 2: Using the Product of the Numbers
We are also given that the product of the two numbers is 108. Mathematically, this is:
\[ ab = 108 \]Step 3: Finding the Difference of Cubes using an Algebraic Identity
We need to find the positive difference between the cubes of these two numbers, which is \(a^3 - b^3\). There is a useful algebraic identity that relates the difference of cubes to the difference and product of the numbers:
\[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]We already know the value of \((a-b)\) and \(ab\). However, we need the value of \((a^2 + b^2)\) to use this identity effectively.
Step 4: Finding the Sum of Squares (\(a^2 + b^2\))
We can find \(a^2 + b^2\) using another algebraic identity involving the difference of the numbers:
\[ (a-b)^2 = a^2 - 2ab + b^2 \]We can rearrange this identity to solve for \(a^2 + b^2\):
\[ a^2 + b^2 = (a-b)^2 + 2ab \]Now, substitute the values we know: \(a-b = 12\) and \(ab = 108\).
\[ a^2 + b^2 = (12)^2 + 2(108) \] \[ a^2 + b^2 = 144 + 216 \] \[ a^2 + b^2 = 360 \]Step 5: Calculating the Difference of Cubes
Now that we have the values for \((a-b)\), \(ab\), and \((a^2 + b^2)\), we can substitute them into the identity for the difference of cubes:
\[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \] \[ a^3 - b^3 = (12)(360 + 108) \] \[ a^3 - b^3 = (12)(468) \]Finally, we multiply 12 by 468:
\[ 12 \times 468 = 5616 \]Thus, the positive difference between the cubes of the two natural numbers is 5616.
Summary of Values
| Expression | Value |
|---|---|
| \( (a-b)^3 \) | 1728 |
| \( a-b \) | 12 |
| \( ab \) | 108 |
| \( a^2 + b^2 \) | 360 |
| \( a^3 - b^3 \) | 5616 |
Final Answer
The positive difference between the cubes of the two given natural numbers is 5616.
Revision Table: Key Concepts
This problem utilizes fundamental algebraic identities. Understanding these is crucial for solving similar problems involving powers of sums or differences of numbers.
| Identity | Formula |
|---|---|
| Difference of Cubes | \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\) |
| Square of a Difference | \((a-b)^2 = a^2 - 2ab + b^2\) |
Additional Information: Natural Numbers and Identities
Natural Numbers: These are the positive integers starting from 1 (1, 2, 3, ...). In this problem, the numbers \(a\) and \(b\) are stated to be natural numbers.
Algebraic Identities: These are equations that are true for all possible values of the variables involved. The identities used in this solution are powerful tools for manipulating and simplifying algebraic expressions, especially when dealing with powers of sums, differences, and products.
By using these identities, we were able to find the required value \(a^3 - b^3\) without needing to individually determine the values of \(a\) and \(b\).
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