Expand Using the Binomial Theorem ( square root of 2x+1/2y)^4

${(\sqrt{2} x + \frac{1}{2} y)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(\sqrt{2} x)}^{4 - k} \cdot {(\frac{1}{2} y)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

Multiply ${(\sqrt{2} x)}^{4}$ by $1$ .

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

Apply the product rule to $\sqrt{2} x$ .

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

Rewrite ${\sqrt{2}}^{4}$ as $2^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $4$ .

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

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

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

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

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

Combine $\frac{1}{2}$ and $y$ .

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

Apply the product rule to $\frac{y}{2}$ .

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

Anything raised to $0$ is $1$ .

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

Anything raised to $0$ is $1$ .

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

Divide $1$ by $1$ .

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

Multiply $4$ by $1$ .

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

Apply the product rule to $\sqrt{2} x$ .

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

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

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

Raise $2$ to the power of $3$ .

$4 x^{4} + 4 \cdot (\sqrt{8} x^{3}) \cdot {(\frac{1}{2} y)}^{1} + 6 \cdot {(\sqrt{2} x)}^{2} \cdot {(\frac{1}{2} y)}^{2} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $2$ by $4$ .

$4 x^{4} + 8 (\sqrt{2} x^{3}) \cdot {(\frac{1}{2} y)}^{1} + 6 \cdot {(\sqrt{2} x)}^{2} \cdot {(\frac{1}{2} y)}^{2} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Combine $\frac{1}{2}$ and $y$ .

$4 x^{4} + 8 \sqrt{2} x^{3} \cdot {(\frac{y}{2})}^{1} + 6 \cdot {(\sqrt{2} x)}^{2} \cdot {(\frac{1}{2} y)}^{2} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Simplify.

$4 x^{4} + 8 \sqrt{2} x^{3} \cdot \frac{y}{2} + 6 \cdot {(\sqrt{2} x)}^{2} \cdot {(\frac{1}{2} y)}^{2} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Cancel the common factor of $2$ .

Factor $2$ out of $8 \sqrt{2} x^{3}$ .

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

Cancel the common factor.

Rewrite the expression.

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

Apply the product rule to $\sqrt{2} x$ .

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $2$ by $6$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 12 x^{2} \cdot {(\frac{1}{2} y)}^{2} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Combine $\frac{1}{2}$ and $y$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 12 x^{2} \cdot {(\frac{y}{2})}^{2} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Apply the product rule to $\frac{y}{2}$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 12 x^{2} \cdot \frac{y^{2}}{2^{2}} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

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

$4 x^{4} + 4 \sqrt{2} x^{3} y + 12 x^{2} \cdot \frac{y^{2}}{4} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Cancel the common factor of $4$ .

Factor $4$ out of $12 x^{2}$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 4 (3 x^{2}) \cdot \frac{y^{2}}{4} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Cancel the common factor.

$4 x^{4} + 4 \sqrt{2} x^{3} y + 4 (3 x^{2}) \cdot \frac{y^{2}}{4} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Rewrite the expression.

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} \cdot y^{2} + 4 \cdot {(\sqrt{2} x)}^{1} \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Simplify.

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + 4 \cdot (\sqrt{2} x) \cdot {(\frac{1}{2} y)}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Combine $\frac{1}{2}$ and $y$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + 4 \sqrt{2} x \cdot {(\frac{y}{2})}^{3} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Apply the product rule to $\frac{y}{2}$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + 4 \sqrt{2} x \cdot \frac{y^{3}}{2^{3}} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Raise $2$ to the power of $3$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + 4 \sqrt{2} x \cdot \frac{y^{3}}{8} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Cancel the common factor of $4$ .

Factor $4$ out of $4 \sqrt{2} x$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + 4 (\sqrt{2} x) \cdot \frac{y^{3}}{8} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Factor $4$ out of $8$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + 4 (\sqrt{2} x) \cdot \frac{y^{3}}{4 (2)} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Cancel the common factor.

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + 4 (\sqrt{2} x) \cdot \frac{y^{3}}{4 \cdot 2} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Rewrite the expression.

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \sqrt{2} x \cdot \frac{y^{3}}{2} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Combine $\sqrt{2}$ and $\frac{y^{3}}{2}$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + x \cdot \frac{\sqrt{2} y^{3}}{2} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Combine $x$ and $\frac{\sqrt{2} y^{3}}{2}$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x (\sqrt{2} y^{3})}{2} + 1 \cdot {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Multiply ${(\sqrt{2} x)}^{0}$ by $1$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + {(\sqrt{2} x)}^{0} \cdot {(\frac{1}{2} y)}^{4}$

Apply the product rule to $\sqrt{2} x$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + {\sqrt{2}}^{0} x^{0} \cdot {(\frac{1}{2} y)}^{4}$

Anything raised to $0$ is $1$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + 1 x^{0} \cdot {(\frac{1}{2} y)}^{4}$

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

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + x^{0} \cdot {(\frac{1}{2} y)}^{4}$

Anything raised to $0$ is $1$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + 1 \cdot {(\frac{1}{2} y)}^{4}$

Multiply ${(\frac{1}{2} y)}^{4}$ by $1$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + {(\frac{1}{2} y)}^{4}$

Combine $\frac{1}{2}$ and $y$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + {(\frac{y}{2})}^{4}$

Apply the product rule to $\frac{y}{2}$ .

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + \frac{y^{4}}{2^{4}}$

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

$4 x^{4} + 4 \sqrt{2} x^{3} y + 3 x^{2} y^{2} + \frac{x \sqrt{2} y^{3}}{2} + \frac{y^{4}}{16}$

Expand Using the Binomial Theorem ( square root of 2x+1/2y)^4

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