If \(a\) and \(b\) are distinct integers, prove that \(a - b\) is a factor of \(a^n - b^n\), whenever \(n\) is a positive integer.
Use expansion: \(a^n = (a - b + b)^n\). Hence \(a^n - b^n\) is divisible by \(a - b\).
Evaluate \((\sqrt{3} + \sqrt{2})^6 - (\sqrt{3} - \sqrt{2})^6\).
396\(\sqrt{6}\)
Find the value of \((a^2 + \sqrt{a^2 - 1})^4 + (a^2 - \sqrt{a^2 - 1})^4\).
\(2a^8 + 12a^6 - 10a^4 - 4a^2 + 2\)
Find an approximation of \((0.99)^5\) using the first three terms of its expansion.
0.9510
Expand using Binomial Theorem: \(\left(1 + \dfrac{x}{2} - \dfrac{2}{x}\right)^4\), \(x \neq 0\).
\(\dfrac{16}{x} + \dfrac{8}{x^2} - \dfrac{32}{x^3} + \dfrac{16}{x^4} - 4x + \dfrac{x^2}{2} + \dfrac{x^3}{2} + \dfrac{x^4}{16} - 5\)
Find the expansion of \((3x^2 - 2ax + 3a^2)^3\) using binomial theorem.
27x^6 - 54ax^5 + 117a^2x^4 - 116a^3x^3 + 117a^4x^2 - 54a^5x + 27a^6