Algebra Examples

$5 (\cos (135 °) + i \sin (135 °)) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 1

Simplify each term.

Tap for more steps...

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.

$5 (- \cos (45) + i \sin (135 °)) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 1.2

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

$5 (- \frac{\sqrt{2}}{2} + i \sin (135 °)) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 1.3

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

$5 (- \frac{\sqrt{2}}{2} + i \sin (45)) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 1.4

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

$5 (- \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 1.5

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

$5 (- \frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2}) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 2

Apply the distributive property.

$(5 (- \frac{\sqrt{2}}{2}) + 5 \frac{i \sqrt{2}}{2}) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 3

Multiply $5 (- \frac{\sqrt{2}}{2})$ .

Tap for more steps...

Step 3.1

Multiply $- 1$ by $5$ .

$(- 5 \frac{\sqrt{2}}{2} + 5 \frac{i \sqrt{2}}{2}) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 3.2

Combine $- 5$ and $\frac{\sqrt{2}}{2}$ .

$(\frac{- 5 \sqrt{2}}{2} + 5 \frac{i \sqrt{2}}{2}) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Combine $5$ and $\frac{i \sqrt{2}}{2}$ .

$(\frac{- 5 \sqrt{2}}{2} + \frac{5 (i \sqrt{2})}{2}) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 4.2

Move the negative in front of the fraction.

$(- \frac{5 \sqrt{2}}{2} + \frac{5 i \sqrt{2}}{2}) \cdot 2 (\cos (45 °) + i \sin (45 °))$

Step 4.3

Simplify terms.

Tap for more steps...

Step 4.3.1

Apply the distributive property.

$(- \frac{5 \sqrt{2}}{2} \cdot 2 + \frac{5 i \sqrt{2}}{2} \cdot 2) (\cos (45 °) + i \sin (45 °))$

Step 4.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.1

Move the leading negative in $- \frac{5 \sqrt{2}}{2}$ into the numerator.

$(\frac{- 5 \sqrt{2}}{2} \cdot 2 + \frac{5 i \sqrt{2}}{2} \cdot 2) (\cos (45 °) + i \sin (45 °))$

Step 4.3.2.2

Cancel the common factor.

$(\frac{- 5 \sqrt{2}}{2} \cdot 2 + \frac{5 i \sqrt{2}}{2} \cdot 2) (\cos (45 °) + i \sin (45 °))$

Step 4.3.2.3

Rewrite the expression.

$(- 5 \sqrt{2} + \frac{5 i \sqrt{2}}{2} \cdot 2) (\cos (45 °) + i \sin (45 °))$

Step 4.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.3.1

Cancel the common factor.

$(- 5 \sqrt{2} + \frac{5 i \sqrt{2}}{2} \cdot 2) (\cos (45 °) + i \sin (45 °))$

Step 4.3.3.2

Rewrite the expression.

$(- 5 \sqrt{2} + 5 i \sqrt{2}) (\cos (45 °) + i \sin (45 °))$

Step 4.4

Simplify each term.

Tap for more steps...

Step 4.4.1

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

$(- 5 \sqrt{2} + 5 i \sqrt{2}) (\frac{\sqrt{2}}{2} + i \sin (45 °))$

Step 4.4.2

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

$(- 5 \sqrt{2} + 5 i \sqrt{2}) (\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})$

Step 4.4.3

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

$(- 5 \sqrt{2} + 5 i \sqrt{2}) (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 5

Expand $(- 5 \sqrt{2} + 5 i \sqrt{2}) (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$ using the FOIL Method.

Tap for more steps...

Step 5.1

Apply the distributive property.

$- 5 \sqrt{2} (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2}) + 5 i \sqrt{2} (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 5.2

Apply the distributive property.

$- 5 \sqrt{2} \frac{\sqrt{2}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 5.3

Apply the distributive property.

$- 5 \sqrt{2} \frac{\sqrt{2}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6

Simplify and combine like terms.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Multiply $- 5 \sqrt{2} \frac{\sqrt{2}}{2}$ .

Tap for more steps...

Step 6.1.1.1

Combine $\frac{\sqrt{2}}{2}$ and $- 5$ .

$\frac{\sqrt{2} \cdot - 5}{2} \sqrt{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.1.2

Combine $\frac{\sqrt{2} \cdot - 5}{2}$ and $\sqrt{2}$ .

$\frac{\sqrt{2} \cdot - 5 \sqrt{2}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.1.3

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

$\frac{- 5 ({\sqrt{2}}^{1} \sqrt{2})}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.1.4

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

$\frac{- 5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.1.5

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

$\frac{- 5 {\sqrt{2}}^{1 + 1}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.1.6

Add $1$ and $1$ .

$\frac{- 5 {\sqrt{2}}^{2}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.2

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

Tap for more steps...

Step 6.1.2.1

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

$\frac{- 5 {(2^{\frac{1}{2}})}^{2}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.2.2

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

$\frac{- 5 \cdot 2^{\frac{1}{2} \cdot 2}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.2.3

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

$\frac{- 5 \cdot 2^{\frac{2}{2}}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.2.4.1

Cancel the common factor.

$\frac{- 5 \cdot 2^{\frac{2}{2}}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.2.4.2

Rewrite the expression.

$\frac{- 5 \cdot 2^{1}}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.2.5

Evaluate the exponent.

$\frac{- 5 \cdot 2}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.3

Multiply $- 5$ by $2$ .

$\frac{- 10}{2} - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.4

Divide $- 10$ by $2$ .

$- 5 - 5 \sqrt{2} \frac{i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.5

Multiply $- 5 \sqrt{2} \frac{i \sqrt{2}}{2}$ .

Tap for more steps...

Step 6.1.5.1

Combine $\frac{i \sqrt{2}}{2}$ and $- 5$ .

$- 5 + \frac{i \sqrt{2} \cdot - 5}{2} \sqrt{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.5.2

Combine $\frac{i \sqrt{2} \cdot - 5}{2}$ and $\sqrt{2}$ .

$- 5 + \frac{i \sqrt{2} \cdot - 5 \sqrt{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.5.3

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

$- 5 + \frac{i \cdot - 5 ({\sqrt{2}}^{1} \sqrt{2})}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.5.4

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

$- 5 + \frac{i \cdot - 5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.5.5

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

$- 5 + \frac{i \cdot - 5 {\sqrt{2}}^{1 + 1}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.5.6

Add $1$ and $1$ .

$- 5 + \frac{i \cdot - 5 {\sqrt{2}}^{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.6

Simplify the numerator.

Tap for more steps...

Step 6.1.6.1

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

Tap for more steps...

Step 6.1.6.1.1

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

$- 5 + \frac{i \cdot - 5 {(2^{\frac{1}{2}})}^{2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.6.1.2

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

$- 5 + \frac{i \cdot - 5 \cdot 2^{\frac{1}{2} \cdot 2}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.6.1.3

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

$- 5 + \frac{i \cdot - 5 \cdot 2^{\frac{2}{2}}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.6.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.6.1.4.1

Cancel the common factor.

$- 5 + \frac{i \cdot - 5 \cdot 2^{\frac{2}{2}}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.6.1.4.2

Rewrite the expression.

$- 5 + \frac{i \cdot - 5 \cdot 2^{1}}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.6.1.5

Evaluate the exponent.

$- 5 + \frac{i \cdot - 5 \cdot 2}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.6.2

Multiply $2$ by $- 5$ .

$- 5 + \frac{i \cdot - 10}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.7

Move $- 10$ to the left of $i$ .

$- 5 + \frac{- 10 i}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.8

Cancel the common factor of $- 10$ and $2$ .

Tap for more steps...

Step 6.1.8.1

Factor $2$ out of $- 10 i$ .

$- 5 + \frac{2 (- 5 i)}{2} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.8.2

Cancel the common factors.

Tap for more steps...

Step 6.1.8.2.1

Factor $2$ out of $2$ .

$- 5 + \frac{2 (- 5 i)}{2 (1)} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.8.2.2

Cancel the common factor.

$- 5 + \frac{2 (- 5 i)}{2 \cdot 1} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.8.2.3

Rewrite the expression.

$- 5 + \frac{- 5 i}{1} + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.8.2.4

Divide $- 5 i$ by $1$ .

$- 5 - 5 i + 5 i \sqrt{2} \frac{\sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.9

Multiply $5 i \sqrt{2} \frac{\sqrt{2}}{2}$ .

Tap for more steps...

Step 6.1.9.1

Combine $\frac{\sqrt{2}}{2}$ and $5$ .

$- 5 - 5 i + \frac{\sqrt{2} \cdot 5}{2} i \sqrt{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.9.2

Combine $\frac{\sqrt{2} \cdot 5}{2}$ and $i$ .

$- 5 - 5 i + \frac{\sqrt{2} \cdot 5 i}{2} \sqrt{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.9.3

Combine $\frac{\sqrt{2} \cdot 5 i}{2}$ and $\sqrt{2}$ .

$- 5 - 5 i + \frac{\sqrt{2} \cdot 5 i \sqrt{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.9.4

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

$- 5 - 5 i + \frac{5 i ({\sqrt{2}}^{1} \sqrt{2})}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.9.5

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

$- 5 - 5 i + \frac{5 i ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.9.6

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

$- 5 - 5 i + \frac{5 i {\sqrt{2}}^{1 + 1}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.9.7

Add $1$ and $1$ .

$- 5 - 5 i + \frac{5 i {\sqrt{2}}^{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.10

Simplify the numerator.

Tap for more steps...

Step 6.1.10.1

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

Tap for more steps...

Step 6.1.10.1.1

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

$- 5 - 5 i + \frac{5 i {(2^{\frac{1}{2}})}^{2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.10.1.2

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

$- 5 - 5 i + \frac{5 i \cdot 2^{\frac{1}{2} \cdot 2}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.10.1.3

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

$- 5 - 5 i + \frac{5 i \cdot 2^{\frac{2}{2}}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.10.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.10.1.4.1

Cancel the common factor.

$- 5 - 5 i + \frac{5 i \cdot 2^{\frac{2}{2}}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.10.1.4.2

Rewrite the expression.

$- 5 - 5 i + \frac{5 i \cdot 2^{1}}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.10.1.5

Evaluate the exponent.

$- 5 - 5 i + \frac{5 i \cdot 2}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.10.2

Multiply $2$ by $5$ .

$- 5 - 5 i + \frac{10 i}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.11

Cancel the common factor of $10$ and $2$ .

Tap for more steps...

Step 6.1.11.1

Factor $2$ out of $10 i$ .

$- 5 - 5 i + \frac{2 (5 i)}{2} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.11.2

Cancel the common factors.

Tap for more steps...

Step 6.1.11.2.1

Factor $2$ out of $2$ .

$- 5 - 5 i + \frac{2 (5 i)}{2 (1)} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.11.2.2

Cancel the common factor.

$- 5 - 5 i + \frac{2 (5 i)}{2 \cdot 1} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.11.2.3

Rewrite the expression.

$- 5 - 5 i + \frac{5 i}{1} + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.11.2.4

Divide $5 i$ by $1$ .

$- 5 - 5 i + 5 i + 5 i \sqrt{2} \frac{i \sqrt{2}}{2}$

Step 6.1.12

Multiply $5 i \sqrt{2} \frac{i \sqrt{2}}{2}$ .

Tap for more steps...

Step 6.1.12.1

Combine $\frac{i \sqrt{2}}{2}$ and $5$ .

$- 5 - 5 i + 5 i + \frac{i \sqrt{2} \cdot 5}{2} i \sqrt{2}$

Step 6.1.12.2

Combine $\frac{i \sqrt{2} \cdot 5}{2}$ and $i$ .

$- 5 - 5 i + 5 i + \frac{i \sqrt{2} \cdot 5 i}{2} \sqrt{2}$

Step 6.1.12.3

Raise $i$ to the power of $1$ .

$- 5 - 5 i + 5 i + \frac{\sqrt{2} \cdot 5 (i^{1} i)}{2} \sqrt{2}$

Step 6.1.12.4

Raise $i$ to the power of $1$ .

$- 5 - 5 i + 5 i + \frac{\sqrt{2} \cdot 5 (i^{1} i^{1})}{2} \sqrt{2}$

Step 6.1.12.5

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

$- 5 - 5 i + 5 i + \frac{\sqrt{2} \cdot 5 i^{1 + 1}}{2} \sqrt{2}$

Step 6.1.12.6

Add $1$ and $1$ .

$- 5 - 5 i + 5 i + \frac{\sqrt{2} \cdot 5 i^{2}}{2} \sqrt{2}$

Step 6.1.12.7

Combine $\frac{\sqrt{2} \cdot 5 i^{2}}{2}$ and $\sqrt{2}$ .

$- 5 - 5 i + 5 i + \frac{\sqrt{2} \cdot 5 i^{2} \sqrt{2}}{2}$

Step 6.1.12.8

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

$- 5 - 5 i + 5 i + \frac{5 i^{2} ({\sqrt{2}}^{1} \sqrt{2})}{2}$

Step 6.1.12.9

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

$- 5 - 5 i + 5 i + \frac{5 i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2}$

Step 6.1.12.10

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

$- 5 - 5 i + 5 i + \frac{5 i^{2} {\sqrt{2}}^{1 + 1}}{2}$

Step 6.1.12.11

Add $1$ and $1$ .

$- 5 - 5 i + 5 i + \frac{5 i^{2} {\sqrt{2}}^{2}}{2}$

Step 6.1.13

Simplify the numerator.

Tap for more steps...

Step 6.1.13.1

Rewrite $i^{2}$ as $- 1$ .

$- 5 - 5 i + 5 i + \frac{5 \cdot - 1 {\sqrt{2}}^{2}}{2}$

Step 6.1.13.2

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

Tap for more steps...

Step 6.1.13.2.1

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

$- 5 - 5 i + 5 i + \frac{5 \cdot - 1 {(2^{\frac{1}{2}})}^{2}}{2}$

Step 6.1.13.2.2

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

$- 5 - 5 i + 5 i + \frac{5 \cdot - 1 \cdot 2^{\frac{1}{2} \cdot 2}}{2}$

Step 6.1.13.2.3

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

$- 5 - 5 i + 5 i + \frac{5 \cdot - 1 \cdot 2^{\frac{2}{2}}}{2}$

Step 6.1.13.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.13.2.4.1

Cancel the common factor.

$- 5 - 5 i + 5 i + \frac{5 \cdot - 1 \cdot 2^{\frac{2}{2}}}{2}$

Step 6.1.13.2.4.2

Rewrite the expression.

$- 5 - 5 i + 5 i + \frac{5 \cdot - 1 \cdot 2^{1}}{2}$

Step 6.1.13.2.5

Evaluate the exponent.

$- 5 - 5 i + 5 i + \frac{5 \cdot - 1 \cdot 2}{2}$

Step 6.1.13.3

Combine exponents.

Tap for more steps...

Step 6.1.13.3.1

Factor out negative.

$- 5 - 5 i + 5 i + \frac{- (5 \cdot 2)}{2}$

Step 6.1.13.3.2

Multiply $5$ by $2$ .

$- 5 - 5 i + 5 i + \frac{- 1 \cdot 10}{2}$

Step 6.1.13.3.3

Multiply $- 1$ by $10$ .

$- 5 - 5 i + 5 i + \frac{- 10}{2}$

Step 6.1.14

Divide $- 10$ by $2$ .

$- 5 - 5 i + 5 i - 5$

Step 6.2

Subtract $5$ from $- 5$ .

$- 10 - 5 i + 5 i$

Step 6.3

Add $- 5 i$ and $5 i$ .

$- 10 + 0$

Step 6.4

Add $- 10$ and $0$ .

$- 10$