Answer:
1680
Step-by-step explanation:
To find the product of the two numbers, you can work out the values of the numbers, or you can work directly with the sum and difference figures.
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We note that the square of a sum is ...
(a +b)² = a² +2ab +b²
and the square of a difference is ...
(a -b)² = a² -2ab +b²
The difference of these squares is ...
(a +b)² -(a -b)² = (a² +2ab +b²) -(a² -2ab +b²) = 4ab
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That is, dividing the difference of the squares of the sum and difference by 4 will give the desired product:
P = (83² -13²)/4 = (6889 -169)/4 = 6720/4 = 1680
The product of the two numbers is 1680.
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Additional comment
If we factor the formula we derived, we find ...
P = ((a +b)² -(a -b)²)/4 = ((a +b) +(a -b))/2 × ((a +b) -(a -b))/2
This simplifies to ...
(2a)/2 × (2b)/2 = a × b
In short:
Using this fact, we find a=(83+13)/2 = 48; b=(83-13)/2 = 35, and the product is (48)(35) = 1680, as above.
