Binomial Theorem

Expand each of the expressions in Exercises 1 to 5.

Expand each of the expressions in Exercises 1 to 5.

Expand each of the expressions in Exercises 1 to 5.

Expand each of the expressions in Exercises 1 to 5.

Expand each of the expressions in Exercises 1 to 5.

Using binomial theorem, evaluate each of the following: (96)^3

Using binomial theorem, evaluate each of the following: (102)^5

Using binomial theorem, evaluate each of the following: (101)^4

Using binomial theorem, evaluate each of the following: (99)^5

Using Binomial Theorem, indicate which number is larger (1.1)^{10000} or 1000.

Find $(a + b)^4 - (a - b)^4$. Hence, evaluate $(\sqrt{3} + \sqrt{2})^4 - (\sqrt{3} - \sqrt{2})^4$.

Find $(x + 1)^6 + (x - 1)^6$. Hence or otherwise evaluate $(\sqrt{2} + 1)^6 + (\sqrt{2} - 1)^6$.

Show that $9^{n+1} - 8n - 9$ is divisible by $64$, whenever $n$ is a positive integer.

Prove that:

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.

Evaluate ($\sqrt{3}$ + $\sqrt{2}$)^6 - ($\sqrt{3}$ - $\sqrt{2}$)^6.

Find the value of ($a^{2}$ + \sqrt{a^{2} - 1})^4 + ($a^{2}$ - \sqrt{a^{2} - 1})^4.

Find an approximation of (0.99)^5 using the first three terms of its expansion.

Expand using Binomial Theorem $\left(1 + \dfrac{x}{2} - \dfrac{2}{x}\right)^4$, $x \neq 0$.

Find the expansion of $(3x^2 - 2ax + 3a^2)^3$ using binomial theorem.