Algebra Examples

$2 \sqrt{2} (\cos (135) + i \sin (135))$

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.

$2 \sqrt{2} (- \cos (45) + i \sin (135))$

Step 1.2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$2 \sqrt{2} (- \frac{\sqrt{2}}{2} + i \sin (135))$

Step 1.3

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

$2 \sqrt{2} (- \frac{\sqrt{2}}{2} + i \sin (45))$

Step 1.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.5

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

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

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

Step 2.2.2

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

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

Step 2.2.3

Cancel the common factor.

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

Step 2.2.4

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

Cancel the common factor of $2$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Add $1$ and $1$ .

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

Step 12

Simplify each term.

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Step 12.1

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

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.1.3

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

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

Step 12.1.4

Cancel the common factor of $2$ .

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Step 12.1.4.1

Cancel the common factor.

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

Step 12.1.4.2

Rewrite the expression.

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

Step 12.1.5

Evaluate the exponent.

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

Step 12.2

Multiply $- 1$ by $2$ .

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

Step 12.3

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

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

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

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

Step 12.3.4

Cancel the common factor of $2$ .

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Step 12.3.4.1

Cancel the common factor.

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

Step 12.3.4.2

Rewrite the expression.

$- 2 + 2^{1} i$

Step 12.3.5

Evaluate the exponent.

$- 2 + 2 i$