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

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

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

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

Evaluate the exponent.

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

Multiply $6$ by $4$ .

$x^{4} + 24 x^{3} + 6 \cdot {(x)}^{2} \cdot {(6)}^{2} + 4 \cdot {(x)}^{1} \cdot {(6)}^{3} + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

Multiply $6$ by ${(6)}^{2}$ by adding the exponents.

Move ${(6)}^{2}$ .

$x^{4} + 24 x^{3} + {(6)}^{2} \cdot 6 \cdot {(x)}^{2} + 4 \cdot {(x)}^{1} \cdot {(6)}^{3} + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

Multiply ${(6)}^{2}$ by $6$ .

Raise $6$ to the power of $1$ .

$x^{4} + 24 x^{3} + {(6)}^{2} \cdot 6^{1} \cdot {(x)}^{2} + 4 \cdot {(x)}^{1} \cdot {(6)}^{3} + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + 24 x^{3} + 6^{2 + 1} \cdot {(x)}^{2} + 4 \cdot {(x)}^{1} \cdot {(6)}^{3} + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

Add $2$ and $1$ .

$x^{4} + 24 x^{3} + 6^{3} \cdot {(x)}^{2} + 4 \cdot {(x)}^{1} \cdot {(6)}^{3} + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

Raise $6$ to the power of $3$ .

$x^{4} + 24 x^{3} + 216 \cdot {(x)}^{2} + 4 \cdot {(x)}^{1} \cdot {(6)}^{3} + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

Simplify.

$x^{4} + 24 x^{3} + 216 x^{2} + 4 \cdot x \cdot {(6)}^{3} + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

Raise $6$ to the power of $3$ .

$x^{4} + 24 x^{3} + 216 x^{2} + 4 x \cdot 216 + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

Multiply $216$ by $4$ .

$x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1 \cdot {(x)}^{0} \cdot {(6)}^{4}$

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

$x^{4} + 24 x^{3} + 216 x^{2} + 864 x + {(x)}^{0} \cdot {(6)}^{4}$

Anything raised to $0$ is $1$ .

$x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1 \cdot {(6)}^{4}$

Multiply ${(6)}^{4}$ by $1$ .

$x^{4} + 24 x^{3} + 216 x^{2} + 864 x + {(6)}^{4}$

Raise $6$ to the power of $4$ .

$x^{4} + 24 x^{3} + 216 x^{2} + 864 x + 1296$

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

Related Questions

Support LCMGCF.com by disabling your adblocker.