Using binomial theorem, evaluate $(96)^3$.

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Accepted Answer

> [!TLDR]
> Express $96$ as $100 - 4$ and expand $(100-4)^3$ using the binomial theorem; the calculation becomes straightforward and gives $\boxed{884736}$.

The binomial theorem is powerful when you need to compute powers of numbers that are close to convenient round numbers. Here, $96$ is just $4$ away from $100$, so we can write $96 = 100 - 4$ and expand $(100-4)^3$ using the theorem.

The binomial theorem states that for any real numbers $a$ and $b$, and positive integer $n$:

> [!FORMULA]
> $$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n}b^n$$

For $n=3$, this becomes:
$$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$

Now let's apply this with $a = 100$ and $b = -4$.

**Step-by-step expansion:**

1. **Identify the terms.** We have $(100 + (-4))^3$, so $a = 100$, $b = -4$, and $n = 3$.

2. **Write out the binomial expansion:**
   $$(100-4)^3 = \binom{3}{0}(100)^3 + \binom{3}{1}(100)^2(-4) + \binom{3}{2}(100)(-4)^2 + \binom{3}{3}(-4)^3$$

3. **Calculate each binomial coefficient:**
   - $\binom{3}{0} = 1$
   - $\binom{3}{1} = 3$
   - $\binom{3}{2} = 3$
   - $\binom{3}{3} = 1$

4. **Compute each term separately:**
   - First term: $1 \cdot (100)^3 = 1000000$
   - Second term: $3 \cdot (100)^2 \cdot (-4) = 3 \cdot 10000 \cdot (-4) = -120000$ …

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 $(96)^3$ .

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 $y$ ? …

Concept and Intuition

Step-by-Step Solution

Q.Using binomial theorem, evaluate (96)3(96)^3(96)3.

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Q.Using binomial theorem, evaluate $(96)^3$ .

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