Algebra Examples

$\sqrt{2} (\cos (\frac{3 π}{4}) + i \sin (\frac{3 π}{4}))$

Step 1

Simplify each term.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\sqrt{2} (- \cos (\frac{π}{4}) + i \sin (\frac{3 π}{4}))$

Step 1.2

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

$\sqrt{2} (- \frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{4}))$

Step 1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

Apply the distributive property.

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

Step 3

Multiply $\sqrt{2} (- \frac{\sqrt{2}}{2})$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 4

Multiply $\sqrt{2} \frac{i \sqrt{2}}{2}$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 5

Simplify each term.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Cancel the common factor of $2$ .

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Step 5.1.4.1

Cancel the common factor.

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

Step 5.1.4.2

Rewrite the expression.

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

Step 5.1.5

Evaluate the exponent.

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

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

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

$- 1 + \frac{2^{\frac{2}{2}} i}{2}$

Step 5.4.4

Cancel the common factor of $2$ .

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Step 5.4.4.1

Cancel the common factor.

$- 1 + \frac{2^{\frac{2}{2}} i}{2}$

Step 5.4.4.2

Rewrite the expression.

$- 1 + \frac{2^{1} i}{2}$

Step 5.4.5

Evaluate the exponent.

$- 1 + \frac{2 i}{2}$

Step 5.5

Cancel the common factor of $2$ .

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Step 5.5.1

Cancel the common factor.

$- 1 + \frac{2 i}{2}$

Step 5.5.2

Divide $i$ by $1$ .

$- 1 + i$