Calculation of a
- Given: a = cube root of √2 + 1 - cube root of √2 - 1
- Simplifying the expression:
a = cube root of (√2 + 1) - cube root of (√2 - 1)
- Using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2):
a = (cube root of √2 - 1)(√2 + 1 + cube root of √2)
- Further simplifying:
a = (√2 - 1)(√2 + 1 + cube root of √2)
a = √2 + 1 - cube root of √2

Calculation of a^3 + 3a - 2
- Given: a = √2 + 1 - cube root of √2
- Now, we need to find a^3 + 3a - 2:
a^3 + 3a - 2 = (√2 + 1 - cube root of √2)^3 + 3(√2 + 1 - cube root of √2) - 2
- Using the formula (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3:
a^3 + 3a - 2 = (√2 + 1 - cube root of √2)(√2 + 1 - cube root of √2)(√2 + 1 - cube root of √2) + 3√2 + 3 - 3 cube root of √2 - 2
- Simplifying further:
a^3 + 3a - 2 = (√2 + 1 - cube root of √2)(√2 - 1) + 3√2 + 3 - 3 cube root of √2 - 2
a^3 + 3a - 2 = √2 - 1 + √2 + 1 - cube root of √2 + 3√2 + 1 - 3 cube root of √2 - 2
- Finally, simplifying the expression:
a^3 + 3a - 2 = 4√2 - 3 cube root of √2 - 2
Therefore, a^3 + 3a - 2 = 4√2 - 3 cube root of √2 - 2.