Basic Math Examples

${(a + \sqrt{2})}^{4}$

Step 1

Use the Binomial Theorem.

$a^{4} + 4 a^{3} \sqrt{2} + 6 a^{2} {\sqrt{2}}^{2} + 4 a {\sqrt{2}}^{3} + {\sqrt{2}}^{4}$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$a^{4} + 4 a^{3} \sqrt{2} + 6 a^{2} {(2^{\frac{1}{2}})}^{2} + 4 a {\sqrt{2}}^{3} + {\sqrt{2}}^{4}$

Step 2.1.2

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

$a^{4} + 4 a^{3} \sqrt{2} + 6 a^{2} \cdot 2^{\frac{1}{2} \cdot 2} + 4 a {\sqrt{2}}^{3} + {\sqrt{2}}^{4}$

Step 2.1.3

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

$a^{4} + 4 a^{3} \sqrt{2} + 6 a^{2} \cdot 2^{\frac{2}{2}} + 4 a {\sqrt{2}}^{3} + {\sqrt{2}}^{4}$

Step 2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.4.1

Cancel the common factor.

$a^{4} + 4 a^{3} \sqrt{2} + 6 a^{2} \cdot 2^{\frac{2}{2}} + 4 a {\sqrt{2}}^{3} + {\sqrt{2}}^{4}$

Step 2.1.4.2

Rewrite the expression.

$a^{4} + 4 a^{3} \sqrt{2} + 6 a^{2} \cdot 2^{1} + 4 a {\sqrt{2}}^{3} + {\sqrt{2}}^{4}$

Step 2.1.5

Evaluate the exponent.

$a^{4} + 4 a^{3} \sqrt{2} + 6 a^{2} \cdot 2 + 4 a {\sqrt{2}}^{3} + {\sqrt{2}}^{4}$

Step 2.2

Multiply $2$ by $6$ .

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 4 a {\sqrt{2}}^{3} + {\sqrt{2}}^{4}$

Step 2.3

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

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 4 a \sqrt{2^{3}} + {\sqrt{2}}^{4}$

Step 2.4

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

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 4 a \sqrt{8} + {\sqrt{2}}^{4}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

Factor $4$ out of $8$ .

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 4 a \sqrt{4 (2)} + {\sqrt{2}}^{4}$

Step 2.5.2

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

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 4 a \sqrt{2^{2} \cdot 2} + {\sqrt{2}}^{4}$

Step 2.6

Pull terms out from under the radical.

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 4 a (2 \sqrt{2}) + {\sqrt{2}}^{4}$

Step 2.7

Multiply $2$ by $4$ .

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + {\sqrt{2}}^{4}$

Step 2.8

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

Tap for more steps...

Step 2.8.1

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

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + {(2^{\frac{1}{2}})}^{4}$

Step 2.8.2

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

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + 2^{\frac{1}{2} \cdot 4}$

Step 2.8.3

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

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + 2^{\frac{4}{2}}$

Step 2.8.4

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

Tap for more steps...

Step 2.8.4.1

Factor $2$ out of $4$ .

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + 2^{\frac{2 \cdot 2}{2}}$

Step 2.8.4.2

Cancel the common factors.

Tap for more steps...

Step 2.8.4.2.1

Factor $2$ out of $2$ .

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + 2^{\frac{2 \cdot 2}{2 (1)}}$

Step 2.8.4.2.2

Cancel the common factor.

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + 2^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 2.8.4.2.3

Rewrite the expression.

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + 2^{\frac{2}{1}}$

Step 2.8.4.2.4

Divide $2$ by $1$ .

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + 2^{2}$

Step 2.9

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

$a^{4} + 4 a^{3} \sqrt{2} + 12 a^{2} + 8 a \sqrt{2} + 4$