Using Binomial theorem evaluate $(99)^5$.

**OR**

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

Question

Using Binomial theorem evaluate $(99)^5$.

**OR**

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

Accepted Answer

>

> [!TLDR]
> Writing $99=100-1$ and expanding $(100-1)^5$ via the Binomial theorem gives $9{,}509{,}900{,}499$.
>
> Write $99 = 100-1$. By the Binomial theorem,
> $$(100-1)^5 = \sum_{k=0}^{5} {}^5C_k\,(100)^{5-k}(-1)^k$$
> $$= {}^5C_0(100)^5 - {}^5C_1(100)^4 + {}^5C_2(100)^3 - {}^5C_3(100)^2 + {}^5C_4(100) - {}^5C_5$$
> $$= (100)^5 - 5(100)^4 + 10(100)^3 - 10(100)^2 + 5(100) - 1$$
>
> Computing each term:
> $(100)^5 = 10{,}000{,}000{,}000$
> $5(100)^4 = 500{,}000{,}000$
> $10(100)^3 = 10{,}000{,}000$
> $10(100)^2 = 100{,}000$
> $5(100) = 500$
>
> Adding with the alternating signs:
> $$10{,}000{,}000{,}000 - 500{,}000{,}000 + 10{,}000{,}000 - 100{,}000 + 500 - 1 = 9{,}509{,}900{,}499$$
> …

Power	Expansion
$(x + y)^0$	$1$
$(x + y)^1$	$x + y$
$(x + y)^2$	$x^2 + 2xy + y^2$
$(x + y)^3$	$x^3 + 3x^2y + 3xy^2 + y^3$
$(x + y)^4$	$x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

$k$	$\binom{4}{k}$	$x^{4-k}$	$y^k$	Term
0	$\binom{4}{0}=1$	$x^4$	$y^0$	$1 \cdot x^4$
1	$\binom{4}{1}=4$	$x^3$	$y^1$	$4x^3y$
2	$\binom{4}{2}=6$	$x^2$	$y^2$	$6x^2y^2$
3	$\binom{4}{3}=4$	$x^1$	$y^3$	$4xy^3$
4	$\binom{4}{4}=1$	$x^0$	$y^4$	$1 \cdot y^4$

Q.Using Binomial theorem evaluate $(99)^5$ . OR Find $(a + b)^4 - (a - b)^4$ . Hence evaluate $\left(\sqrt{3} + \sqrt{2}\right)^4 - \left(\sqrt{3} - \sqrt{2}\right)^4$ .

Concept understanding — Binomial Theorem Expansion

The Problem That Started It All

The Pattern You Already Know

The Precise Statement

Where Do Those Coefficients Come From?

The General Term

What If It's Not Just $x$ and $y$ ? …

Q.Using Binomial theorem evaluate (99)5(99)^5(99)5. OR Find (a+b)4−(a−b)4(a + b)^4 - (a - b)^4(a+b)4−(a−b)4. Hence evaluate (3+2)4−(3−2)4\left(\sqrt{3} + \sqrt{2}\right)^4 - \left(\sqrt{3} - \sqrt{2}\right)^4(3​+2​)4−(3​−2​)4.

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Q.Using Binomial theorem evaluate $(99)^5$ . OR Find $(a + b)^4 - (a - b)^4$ . Hence evaluate $\left(\sqrt{3} + \sqrt{2}\right)^4 - \left(\sqrt{3} - \sqrt{2}\right)^4$ .

Unlock everything free for 14 days