Expand Using the Binomial Theorem (x+7)^4

${(x + 7)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(x)}^{4 - k} \cdot {(7)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} {(x)}^{4 - 0} \cdot {(7)}^{0} + \frac{4!}{(4 - 1)! 1!} {(x)}^{4 - 1} \cdot {(7)}^{1} + \frac{4!}{(4 - 2)! 2!} {(x)}^{4 - 2} \cdot {(7)}^{2} + \frac{4!}{(4 - 3)! 3!} {(x)}^{4 - 3} \cdot {(7)}^{3} + \frac{4!}{(4 - 4)! 4!} {(x)}^{4 - 4} \cdot {(7)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{4} \cdot {(7)}^{0} + 4 \cdot {(x)}^{3} \cdot {(7)}^{1} + 6 \cdot {(x)}^{2} \cdot {(7)}^{2} + 4 \cdot {(x)}^{1} \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Step 4

Simplify each term.

Multiply ${(x)}^{4}$ by $1$ .

${(x)}^{4} \cdot {(7)}^{0} + 4 \cdot {(x)}^{3} \cdot {(7)}^{1} + 6 \cdot {(x)}^{2} \cdot {(7)}^{2} + 4 \cdot {(x)}^{1} \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Anything raised to $0$ is $1$ .

$x^{4} \cdot 1 + 4 \cdot {(x)}^{3} \cdot {(7)}^{1} + 6 \cdot {(x)}^{2} \cdot {(7)}^{2} + 4 \cdot {(x)}^{1} \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Multiply $x^{4}$ by $1$ .

$x^{4} + 4 \cdot {(x)}^{3} \cdot {(7)}^{1} + 6 \cdot {(x)}^{2} \cdot {(7)}^{2} + 4 \cdot {(x)}^{1} \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Evaluate the exponent.

$x^{4} + 4 x^{3} \cdot 7 + 6 \cdot {(x)}^{2} \cdot {(7)}^{2} + 4 \cdot {(x)}^{1} \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Multiply $7$ by $4$ .

$x^{4} + 28 x^{3} + 6 \cdot {(x)}^{2} \cdot {(7)}^{2} + 4 \cdot {(x)}^{1} \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Raise $7$ to the power of $2$ .

$x^{4} + 28 x^{3} + 6 x^{2} \cdot 49 + 4 \cdot {(x)}^{1} \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Multiply $49$ by $6$ .

$x^{4} + 28 x^{3} + 294 x^{2} + 4 \cdot {(x)}^{1} \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Simplify.

$x^{4} + 28 x^{3} + 294 x^{2} + 4 \cdot x \cdot {(7)}^{3} + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Raise $7$ to the power of $3$ .

$x^{4} + 28 x^{3} + 294 x^{2} + 4 x \cdot 343 + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Multiply $343$ by $4$ .

$x^{4} + 28 x^{3} + 294 x^{2} + 1372 x + 1 \cdot {(x)}^{0} \cdot {(7)}^{4}$

Multiply ${(x)}^{0}$ by $1$ .

$x^{4} + 28 x^{3} + 294 x^{2} + 1372 x + {(x)}^{0} \cdot {(7)}^{4}$

Anything raised to $0$ is $1$ .

$x^{4} + 28 x^{3} + 294 x^{2} + 1372 x + 1 \cdot {(7)}^{4}$

Multiply ${(7)}^{4}$ by $1$ .

$x^{4} + 28 x^{3} + 294 x^{2} + 1372 x + {(7)}^{4}$

Raise $7$ to the power of $4$ .

$x^{4} + 28 x^{3} + 294 x^{2} + 1372 x + 2401$

Expand Using the Binomial Theorem (x+7)^4

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