Answer
The product of the two numbers 65^{1000}-8^{2001}+3^{177}, 79^{1212}-9^{2399}+2^{2001} and 24^{4493}-5^{8192}+7^{1777} is a non- negative.
Work Step by Step
the step by step procedure proving that the above-given statement is true is given below:
A value is nonnegative if it is either positive or zero.
If one of the terms is zero, then their product is also zero and also non-negative .it is then safe to assume all three values are nonzero (since we know the statement is true when one of the values is zero).
If a value is nonzero, then the value is either positive or negative. thus each of the three numbers is either positive or negative.
Then we know that at least two of the three numbers have the same sign (since each number only has the option: positive or negative).
The product of two positive numbers is positive, and the product of two negative numbers is positive. Thus we then know the product of the two numbers with the same sign is positive and thus also nonnegative. Thus we have then shown that the product of two of three numbers is nonnegative.
Note: This is generally true for any three real numbers
