Final answer:
To find the product of two numbers given their sum and the sum of their cubes, we can solve a system of equations. The product of the two numbers is 6.
Step-by-step explanation:
To find the product of the two numbers, we can use algebraic equations. Let's assume the two numbers are x and y. We are given that the sum of the two numbers is 7, so we can write the equation x + y = 7.
We are also given that the sum of their cubes is 217, so we can write the equation x^3 + y^3 = 217. Now, using a property of cubes, we can rewrite the second equation as (x + y)(x^2 - xy + y^2) = 217.
Substituting the value of x + y from the first equation into the second equation, we get 7(x^2 - xy + y^2) = 217. Simplifying this equation, we obtain x^2 - xy + y^2 = 31.
Now, notice that (x + y)^2 = x^2 + 2xy + y^2. Since we know that x + y = 7, we can substitute this value into the equation to get 7^2 = x^2 + 2xy + y^2.
Expanding this equation, we get 49 = x^2 + 2xy + y^2. Comparing this equation to the previous equation x^2 - xy + y^2 = 31, we can subtract the two equations to eliminate the terms containing xy. This gives us 18 = 3xy, or xy = 6.
Therefore, the product of the two numbers is 6.
Learn more about Product of Two Numbers