Algebra Examples

$\sqrt{32} (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.1

Factor $16$ out of $32$ .

$\sqrt{16 (2)} (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.2

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 2} (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 2

Pull terms out from under the radical.

$4 \sqrt{2} (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3

Simplify each term.

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Step 3.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$4 \sqrt{2} (\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4}))$

Step 3.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$4 \sqrt{2} (\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})$

Step 3.3

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$4 \sqrt{2} (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 4

Simplify terms.

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Step 4.1

Apply the distributive property.

$4 \sqrt{2} \frac{\sqrt{2}}{2} + 4 \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

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

$2 (2 \sqrt{2}) \frac{\sqrt{2}}{2} + 4 \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 4.2.2

Cancel the common factor.

$2 (2 \sqrt{2}) \frac{\sqrt{2}}{2} + 4 \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 4.2.3

Rewrite the expression.

$2 \sqrt{2} \sqrt{2} + 4 \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 5

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 6

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7

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

$2 {\sqrt{2}}^{1 + 1} + 4 \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 8

Add $1$ and $1$ .

$2 {\sqrt{2}}^{2} + 4 \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

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

$2 {\sqrt{2}}^{2} + 2 (2 \sqrt{2}) \frac{i \sqrt{2}}{2}$

Step 9.2

Cancel the common factor.

$2 {\sqrt{2}}^{2} + 2 (2 \sqrt{2}) \frac{i \sqrt{2}}{2}$

Step 9.3

Rewrite the expression.

$2 {\sqrt{2}}^{2} + 2 \sqrt{2} (i \sqrt{2})$

Step 10

Raise $\sqrt{2}$ to the power of $1$ .

$2 {\sqrt{2}}^{2} + 2 ({\sqrt{2}}^{1} \sqrt{2}) i$

Step 11

Raise $\sqrt{2}$ to the power of $1$ .

$2 {\sqrt{2}}^{2} + 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) i$

Step 12

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

$2 {\sqrt{2}}^{2} + 2 {\sqrt{2}}^{1 + 1} i$

Step 13

Add $1$ and $1$ .

$2 {\sqrt{2}}^{2} + 2 {\sqrt{2}}^{2} i$

Step 14

Simplify each term.

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Step 14.1

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

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Step 14.1.1

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

$2 {(2^{\frac{1}{2}})}^{2} + 2 {\sqrt{2}}^{2} i$

Step 14.1.2

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

$2 \cdot 2^{\frac{1}{2} \cdot 2} + 2 {\sqrt{2}}^{2} i$

Step 14.1.3

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

$2 \cdot 2^{\frac{2}{2}} + 2 {\sqrt{2}}^{2} i$

Step 14.1.4

Cancel the common factor of $2$ .

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Step 14.1.4.1

Cancel the common factor.

$2 \cdot 2^{\frac{2}{2}} + 2 {\sqrt{2}}^{2} i$

Step 14.1.4.2

Rewrite the expression.

$2 \cdot 2^{1} + 2 {\sqrt{2}}^{2} i$

Step 14.1.5

Evaluate the exponent.

$2 \cdot 2 + 2 {\sqrt{2}}^{2} i$

Step 14.2

Multiply $2$ by $2$ .

$4 + 2 {\sqrt{2}}^{2} i$

Step 14.3

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

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Step 14.3.1

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

$4 + 2 {(2^{\frac{1}{2}})}^{2} i$

Step 14.3.2

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

$4 + 2 \cdot 2^{\frac{1}{2} \cdot 2} i$

Step 14.3.3

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

$4 + 2 \cdot 2^{\frac{2}{2}} i$

Step 14.3.4

Cancel the common factor of $2$ .

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Step 14.3.4.1

Cancel the common factor.

$4 + 2 \cdot 2^{\frac{2}{2}} i$

Step 14.3.4.2

Rewrite the expression.

$4 + 2 \cdot 2^{1} i$

Step 14.3.5

Evaluate the exponent.

$4 + 2 \cdot 2 i$

Step 14.4

Multiply $2$ by $2$ .

$4 + 4 i$