Expand the Trigonometric Expression (4u+1)^4

${(4 u + 1)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $4 u$ .

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

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

$256 u^{4} + 4 {(4 u)}^{3} \cdot 1 + 6 {(4 u)}^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Apply the product rule to $4 u$ .

$256 u^{4} + 4 (4^{3} u^{3}) \cdot 1 + 6 {(4 u)}^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Multiply $4$ by $4^{3}$ by adding the exponents.

Move $4^{3}$ .

$256 u^{4} + 4^{3} \cdot 4 u^{3} \cdot 1 + 6 {(4 u)}^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Multiply $4^{3}$ by $4$ .

Raise $4$ to the power of $1$ .

$256 u^{4} + 4^{3} \cdot 4^{1} u^{3} \cdot 1 + 6 {(4 u)}^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

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

$256 u^{4} + 4^{3 + 1} u^{3} \cdot 1 + 6 {(4 u)}^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Add $3$ and $1$ .

$256 u^{4} + 4^{4} u^{3} \cdot 1 + 6 {(4 u)}^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Simplify $4^{4} u^{3} \cdot 1$ .

$256 u^{4} + 4^{4} u^{3} + 6 {(4 u)}^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

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

$256 u^{4} + 256 u^{3} + 6 {(4 u)}^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Apply the product rule to $4 u$ .

$256 u^{4} + 256 u^{3} + 6 (4^{2} u^{2}) \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Raise $4$ to the power of $2$ .

$256 u^{4} + 256 u^{3} + 6 (16 u^{2}) \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Multiply $16$ by $6$ .

$256 u^{4} + 256 u^{3} + 96 u^{2} \cdot 1^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

One to any power is one.

$256 u^{4} + 256 u^{3} + 96 u^{2} \cdot 1 + 4 (4 u) \cdot 1^{3} + 1^{4}$

Multiply $96$ by $1$ .

$256 u^{4} + 256 u^{3} + 96 u^{2} + 4 (4 u) \cdot 1^{3} + 1^{4}$

Multiply $4$ by $4$ .

$256 u^{4} + 256 u^{3} + 96 u^{2} + 16 u \cdot 1^{3} + 1^{4}$

One to any power is one.

$256 u^{4} + 256 u^{3} + 96 u^{2} + 16 u \cdot 1 + 1^{4}$

Multiply $16$ by $1$ .

$256 u^{4} + 256 u^{3} + 96 u^{2} + 16 u + 1^{4}$

One to any power is one.

$256 u^{4} + 256 u^{3} + 96 u^{2} + 16 u + 1$