i am stuck.
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 use the fact that the sum, difference, product, and division of rational numbers are also rational numbers.
Let's start by expanding the expression using the formula for the cube of a binomial:
(a + √b)^3 = a^3 + 3a^2√b + 3ab + b√b
(a - √b)^3 = a^3 - 3a^2√b + 3ab - b√b
Now let's add the two expanded expressions:
(a + √b)^3 + (a - √b)^3 = (a^3 + 3a^2√b + 3ab + b√b) + (a^3 - 3a^2√b + 3ab - b√b)
By rearranging and regrouping the terms, we can see that the terms involving the square root (√b) cancel out:
= (a^3 + a^3) + (3a^2√b - 3a^2√b) + (3ab + 3ab) + (b√b - b√b)
This simplifies to:
= 2a^3 + 6ab
Since a^3 and ab are both rational numbers (since a is rational) and the sum and product of rational numbers are also rational, we conclude that 2a^3 + 6ab is rational.
Therefore, (a + √b)^3 + (a - √b)^3 is rational when a and b are rational numbers.