Anyone?
Prove that if a and b are rational numbers, (a + ^b)3 + (a - ^b)3 is also rational.
^ = sq root sign
Thanks
To prove that the expression (a + √b)^3 + (a - √b)^3 is rational when a and b are rational numbers, we can simplify the expression and show that it can be expressed as a ratio of two integers.
Let's simplify the given expression:
(a + √b)^3 + (a - √b)^3
Using the binomial formula for (a + √b)^3:
(a + √b)^3 = a^3 + 3a^2√b + 3ab + b√b
Similarly, using the binomial formula for (a - √b)^3:
(a - √b)^3 = a^3 - 3a^2√b + 3ab - b√b
Now, let's add both expressions together:
(a + √b)^3 + (a - √b)^3 = a^3 + 3a^2√b + 3ab + b√b + a^3 - 3a^2√b + 3ab - b√b
Combining like terms:
(a + √b)^3 + (a - √b)^3 = 2a^3 + 6ab
Since a and b are rational numbers, the product and sum of rational numbers is also rational. Therefore, 2a^3 + 6ab is a rational number.
Hence, we have proved that if a and b are rational numbers, (a + √b)^3 + (a - √b)^3 is also a rational number.