Expand Using the Binomial Theorem (1+4x)^6

${(1 + 4 x)}^{6}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{6} \frac{6!}{(6 - k)! k!} \cdot {(1)}^{6 - k} \cdot {(4 x)}^{k}$

Step 2

Expand the summation.

$\frac{6!}{(6 - 0)! 0!} \cdot {(1)}^{6 - 0} \cdot {(4 x)}^{0} + \frac{6!}{(6 - 1)! 1!} \cdot {(1)}^{6 - 1} \cdot {(4 x)}^{1} + \dots + \frac{6!}{(6 - 6)! 6!} \cdot {(1)}^{6 - 6} \cdot {(4 x)}^{6}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(1)}^{6} \cdot {(4 x)}^{0} + 6 \cdot {(1)}^{5} \cdot {(4 x)}^{1} + \dots + 1 \cdot {(1)}^{0} \cdot {(4 x)}^{6}$

Step 4

Simplify the polynomial result.

$1 + 24 x + 240 x^{2} + 1280 x^{3} + 3840 x^{4} + 6144 x^{5} + 4096 x^{6}$

Expand Using the Binomial Theorem (1+4x)^6

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