Trigonometry Examples

${(\sqrt{2} (\cos (\frac{7 π}{6}) + i \sin (\frac{7 π}{6})))}^{4}$

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 third quadrant.

${(\sqrt{2} (- \cos (\frac{π}{6}) + i \sin (\frac{7 π}{6})))}^{4}$

Step 1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

${(\sqrt{2} (- \frac{\sqrt{3}}{2} + i \sin (\frac{7 π}{6})))}^{4}$

Step 1.3

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

${(\sqrt{2} (- \frac{\sqrt{3}}{2} + i (- \sin (\frac{π}{6}))))}^{4}$

Step 1.4

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

${(\sqrt{2} (- \frac{\sqrt{3}}{2} + i (- \frac{1}{2})))}^{4}$

Step 1.5

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

${(\sqrt{2} (- \frac{\sqrt{3}}{2} - \frac{i}{2}))}^{4}$

Step 2

Apply the distributive property.

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

Step 3

Multiply $\sqrt{2} (- \frac{\sqrt{3}}{2})$ .

Tap for more steps...

Step 3.1

Combine $\sqrt{2}$ and $\frac{\sqrt{3}}{2}$ .

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

Step 3.2

Combine using the product rule for radicals.

${(- \frac{\sqrt{2 \cdot 3}}{2} + \sqrt{2} (- \frac{i}{2}))}^{4}$

Step 3.3

Multiply $2$ by $3$ .

${(- \frac{\sqrt{6}}{2} + \sqrt{2} (- \frac{i}{2}))}^{4}$

Step 4

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

${(- \frac{\sqrt{6}}{2} - \frac{\sqrt{2} i}{2})}^{4}$

Step 5

Use the Binomial Theorem.

${(- \frac{\sqrt{6}}{2})}^{4} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6

Simplify terms.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.1.1.1

Apply the product rule to $- \frac{\sqrt{6}}{2}$ .

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

Step 6.1.1.2

Apply the product rule to $\frac{\sqrt{6}}{2}$ .

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

Step 6.1.2

Raise $- 1$ to the power of $4$ .

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

Step 6.1.3

Multiply $\frac{{\sqrt{6}}^{4}}{2^{4}}$ by $1$ .

$\frac{{\sqrt{6}}^{4}}{2^{4}} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.4

Simplify the numerator.

Tap for more steps...

Step 6.1.4.1

Rewrite ${\sqrt{6}}^{4}$ as $6^{2}$ .

Tap for more steps...

Step 6.1.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 6.1.4.1.2

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

$\frac{6^{\frac{1}{2} \cdot 4}}{2^{4}} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.4.1.3

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

$\frac{6^{\frac{4}{2}}}{2^{4}} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.4.1.4

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

Tap for more steps...

Step 6.1.4.1.4.1

Factor $2$ out of $4$ .

$\frac{6^{\frac{2 \cdot 2}{2}}}{2^{4}} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.4.1.4.2

Cancel the common factors.

Tap for more steps...

Step 6.1.4.1.4.2.1

Factor $2$ out of $2$ .

$\frac{6^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.4.1.4.2.2

Cancel the common factor.

$\frac{6^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.4.1.4.2.3

Rewrite the expression.

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

Step 6.1.4.1.4.2.4

Divide $2$ by $1$ .

$\frac{6^{2}}{2^{4}} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.4.2

Raise $6$ to the power of $2$ .

$\frac{36}{2^{4}} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.5

Raise $2$ to the power of $4$ .

$\frac{36}{16} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.6

Cancel the common factor of $36$ and $16$ .

Tap for more steps...

Step 6.1.6.1

Factor $4$ out of $36$ .

$\frac{4 (9)}{16} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.6.2

Cancel the common factors.

Tap for more steps...

Step 6.1.6.2.1

Factor $4$ out of $16$ .

$\frac{4 \cdot 9}{4 \cdot 4} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.6.2.2

Cancel the common factor.

Step 6.1.6.2.3

Rewrite the expression.

$\frac{9}{4} + 4 {(- \frac{\sqrt{6}}{2})}^{3} (- \frac{\sqrt{2} i}{2}) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.7

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.7.1

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

$\frac{9}{4} + 4 {(- \frac{\sqrt{6}}{2})}^{3} \frac{- \sqrt{2} i}{2} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.7.2

Factor $2$ out of $4 {(- \frac{\sqrt{6}}{2})}^{3}$ .

$\frac{9}{4} + 2 (2 {(- \frac{\sqrt{6}}{2})}^{3}) \frac{- \sqrt{2} i}{2} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.7.3

Cancel the common factor.

Step 6.1.7.4

Rewrite the expression.

$\frac{9}{4} + 2 {(- \frac{\sqrt{6}}{2})}^{3} (- \sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.8

Multiply $- 1$ by $2$ .

$\frac{9}{4} - 2 {(- \frac{\sqrt{6}}{2})}^{3} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.1.9.1

Apply the product rule to $- \frac{\sqrt{6}}{2}$ .

$\frac{9}{4} - 2 ({(- 1)}^{3} {(\frac{\sqrt{6}}{2})}^{3}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.9.2

Apply the product rule to $\frac{\sqrt{6}}{2}$ .

$\frac{9}{4} - 2 ({(- 1)}^{3} \frac{{\sqrt{6}}^{3}}{2^{3}}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.10

Raise $- 1$ to the power of $3$ .

$\frac{9}{4} - 2 (- \frac{{\sqrt{6}}^{3}}{2^{3}}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.11

Simplify the numerator.

Tap for more steps...

Step 6.1.11.1

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$\frac{9}{4} - 2 (- \frac{\sqrt{6^{3}}}{2^{3}}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.11.2

Raise $6$ to the power of $3$ .

$\frac{9}{4} - 2 (- \frac{\sqrt{216}}{2^{3}}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.11.3

Rewrite $216$ as $6^{2} \cdot 6$ .

Tap for more steps...

Step 6.1.11.3.1

Factor $36$ out of $216$ .

$\frac{9}{4} - 2 (- \frac{\sqrt{36 (6)}}{2^{3}}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.11.3.2

Rewrite $36$ as $6^{2}$ .

$\frac{9}{4} - 2 (- \frac{\sqrt{6^{2} \cdot 6}}{2^{3}}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.11.4

Pull terms out from under the radical.

$\frac{9}{4} - 2 (- \frac{6 \sqrt{6}}{2^{3}}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.12

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

$\frac{9}{4} - 2 (- \frac{6 \sqrt{6}}{8}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.13

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.13.1

Move the leading negative in $- \frac{6 \sqrt{6}}{8}$ into the numerator.

$\frac{9}{4} - 2 \frac{- 6 \sqrt{6}}{8} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.13.2

Factor $2$ out of $- 2$ .

$\frac{9}{4} + 2 (- 1) \frac{- 6 \sqrt{6}}{8} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.13.3

Factor $2$ out of $8$ .

$\frac{9}{4} + 2 \cdot - 1 \frac{- 6 \sqrt{6}}{2 \cdot 4} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.13.4

Cancel the common factor.

Step 6.1.13.5

Rewrite the expression.

$\frac{9}{4} - 1 \frac{- 6 \sqrt{6}}{4} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.14

Cancel the common factor of $- 6$ and $4$ .

Tap for more steps...

Step 6.1.14.1

Factor $2$ out of $- 6 \sqrt{6}$ .

$\frac{9}{4} - 1 \frac{2 (- 3 \sqrt{6})}{4} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.14.2

Cancel the common factors.

Tap for more steps...

Step 6.1.14.2.1

Factor $2$ out of $4$ .

$\frac{9}{4} - 1 \frac{2 (- 3 \sqrt{6})}{2 (2)} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.14.2.2

Cancel the common factor.

$\frac{9}{4} - 1 \frac{2 (- 3 \sqrt{6})}{2 \cdot 2} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.14.2.3

Rewrite the expression.

$\frac{9}{4} - 1 \frac{- 3 \sqrt{6}}{2} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.15

Move the negative in front of the fraction.

$\frac{9}{4} - 1 (- \frac{3 \sqrt{6}}{2}) (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.16

Multiply $- 1 (- \frac{3 \sqrt{6}}{2})$ .

Tap for more steps...

Step 6.1.16.1

Multiply $- 1$ by $- 1$ .

$\frac{9}{4} + 1 \frac{3 \sqrt{6}}{2} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.16.2

Multiply $\frac{3 \sqrt{6}}{2}$ by $1$ .

$\frac{9}{4} + \frac{3 \sqrt{6}}{2} (\sqrt{2} i) + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.17

Multiply $\frac{3 \sqrt{6}}{2} (\sqrt{2} i)$ .

Tap for more steps...

Step 6.1.17.1

Combine $\sqrt{2}$ and $\frac{3 \sqrt{6}}{2}$ .

$\frac{9}{4} + \frac{\sqrt{2} (3 \sqrt{6})}{2} i + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.17.2

Combine using the product rule for radicals.

$\frac{9}{4} + \frac{3 \sqrt{2 \cdot 6}}{2} i + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.17.3

Multiply $2$ by $6$ .

$\frac{9}{4} + \frac{3 \sqrt{12}}{2} i + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.17.4

Combine $\frac{3 \sqrt{12}}{2}$ and $i$ .

$\frac{9}{4} + \frac{3 \sqrt{12} i}{2} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.18

Simplify the numerator.

Tap for more steps...

Step 6.1.18.1

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 6.1.18.1.1

Factor $4$ out of $12$ .

$\frac{9}{4} + \frac{3 \sqrt{4 (3)} i}{2} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.18.1.2

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

$\frac{9}{4} + \frac{3 \sqrt{2^{2} \cdot 3} i}{2} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.18.2

Pull terms out from under the radical.

$\frac{9}{4} + \frac{3 \cdot 2 \sqrt{3} i}{2} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.18.3

Multiply $3$ by $2$ .

$\frac{9}{4} + \frac{6 \sqrt{3} i}{2} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.19

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 6.1.19.1

Factor $2$ out of $6 \sqrt{3} i$ .

$\frac{9}{4} + \frac{2 (3 \sqrt{3} i)}{2} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.19.2

Cancel the common factors.

Tap for more steps...

Step 6.1.19.2.1

Factor $2$ out of $2$ .

$\frac{9}{4} + \frac{2 (3 \sqrt{3} i)}{2 (1)} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.19.2.2

Cancel the common factor.

$\frac{9}{4} + \frac{2 (3 \sqrt{3} i)}{2 \cdot 1} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.19.2.3

Rewrite the expression.

$\frac{9}{4} + \frac{3 \sqrt{3} i}{1} + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.19.2.4

Divide $3 \sqrt{3} i$ by $1$ .

$\frac{9}{4} + 3 \sqrt{3} i + 6 {(- \frac{\sqrt{6}}{2})}^{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.20

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.1.20.1

Apply the product rule to $- \frac{\sqrt{6}}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i + 6 ({(- 1)}^{2} {(\frac{\sqrt{6}}{2})}^{2}) {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.20.2

Apply the product rule to $\frac{\sqrt{6}}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i + 6 ({(- 1)}^{2} \frac{{\sqrt{6}}^{2}}{2^{2}}) {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.21

Raise $- 1$ to the power of $2$ .

$\frac{9}{4} + 3 \sqrt{3} i + 6 (1 \frac{{\sqrt{6}}^{2}}{2^{2}}) {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.22

Multiply $\frac{{\sqrt{6}}^{2}}{2^{2}}$ by $1$ .

$\frac{9}{4} + 3 \sqrt{3} i + 6 \frac{{\sqrt{6}}^{2}}{2^{2}} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.23

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 6.1.23.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{9}{4} + 3 \sqrt{3} i + 6 \frac{{(6^{\frac{1}{2}})}^{2}}{2^{2}} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.23.2

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

$\frac{9}{4} + 3 \sqrt{3} i + 6 \frac{6^{\frac{1}{2} \cdot 2}}{2^{2}} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.23.3

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

$\frac{9}{4} + 3 \sqrt{3} i + 6 \frac{6^{\frac{2}{2}}}{2^{2}} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.23.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.23.4.1

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i + 6 \frac{6^{\frac{2}{2}}}{2^{2}} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.23.4.2

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i + 6 \frac{6^{1}}{2^{2}} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.23.5

Evaluate the exponent.

$\frac{9}{4} + 3 \sqrt{3} i + 6 \frac{6}{2^{2}} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.24

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

$\frac{9}{4} + 3 \sqrt{3} i + 6 (\frac{6}{4}) {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.25

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.25.1

Factor $2$ out of $6$ .

$\frac{9}{4} + 3 \sqrt{3} i + 2 (3) \frac{6}{4} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.25.2

Factor $2$ out of $4$ .

$\frac{9}{4} + 3 \sqrt{3} i + 2 \cdot 3 \frac{6}{2 \cdot 2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.25.3

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i + 2 \cdot 3 \frac{6}{2 \cdot 2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.25.4

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i + 3 (\frac{6}{2}) {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.26

Combine $3$ and $\frac{6}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i + \frac{3 \cdot 6}{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.27

Multiply $3$ by $6$ .

$\frac{9}{4} + 3 \sqrt{3} i + \frac{18}{2} {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.28

Divide $18$ by $2$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 {(- \frac{\sqrt{2} i}{2})}^{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.29

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.1.29.1

Apply the product rule to $- \frac{\sqrt{2} i}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 ({(- 1)}^{2} {(\frac{\sqrt{2} i}{2})}^{2}) + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.29.2

Apply the product rule to $\frac{\sqrt{2} i}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 ({(- 1)}^{2} \frac{{(\sqrt{2} i)}^{2}}{2^{2}}) + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.29.3

Apply the product rule to $\sqrt{2} i$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 ({(- 1)}^{2} \frac{{\sqrt{2}}^{2} i^{2}}{2^{2}}) + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.30

Raise $- 1$ to the power of $2$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 (1 \frac{{\sqrt{2}}^{2} i^{2}}{2^{2}}) + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.31

Multiply $\frac{{\sqrt{2}}^{2} i^{2}}{2^{2}}$ by $1$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{{\sqrt{2}}^{2} i^{2}}{2^{2}} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.32

Simplify the numerator.

Tap for more steps...

Step 6.1.32.1

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

Tap for more steps...

Step 6.1.32.1.1

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

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{{(2^{\frac{1}{2}})}^{2} i^{2}}{2^{2}} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.32.1.2

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

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2^{\frac{1}{2} \cdot 2} i^{2}}{2^{2}} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.32.1.3

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

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2^{\frac{2}{2}} i^{2}}{2^{2}} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.32.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.32.1.4.1

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2^{\frac{2}{2}} i^{2}}{2^{2}} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.32.1.4.2

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2^{1} i^{2}}{2^{2}} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.32.1.5

Evaluate the exponent.

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2 i^{2}}{2^{2}} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.32.2

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

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2 \cdot - 1}{2^{2}} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.33

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

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2 \cdot - 1}{4} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.34

Multiply $2$ by $- 1$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 (\frac{- 2}{4}) + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.35

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

Tap for more steps...

Step 6.1.35.1

Factor $2$ out of $- 2$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2 (- 1)}{4} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.35.2

Cancel the common factors.

Tap for more steps...

Step 6.1.35.2.1

Factor $2$ out of $4$ .

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2 \cdot - 1}{2 \cdot 2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.35.2.2

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i + 9 \frac{2 \cdot - 1}{2 \cdot 2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.35.2.3

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i + 9 (\frac{- 1}{2}) + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.36

Move the negative in front of the fraction.

$\frac{9}{4} + 3 \sqrt{3} i + 9 (- \frac{1}{2}) + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.37

Multiply $9 (- \frac{1}{2})$ .

Tap for more steps...

Step 6.1.37.1

Multiply $- 1$ by $9$ .

$\frac{9}{4} + 3 \sqrt{3} i - 9 (\frac{1}{2}) + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.37.2

Combine $- 9$ and $\frac{1}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i + \frac{- 9}{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.38

Move the negative in front of the fraction.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} + 4 (- \frac{\sqrt{6}}{2}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.39

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.39.1

Move the leading negative in $- \frac{\sqrt{6}}{2}$ into the numerator.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} + 4 \frac{- \sqrt{6}}{2} {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.39.2

Factor $2$ out of $4$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} + 2 (2) \frac{- \sqrt{6}}{2} {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.39.3

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} + 2 \cdot 2 \frac{- \sqrt{6}}{2} {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.39.4

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} + 2 (- \sqrt{6}) {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.40

Multiply $- 1$ by $2$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} {(- \frac{\sqrt{2} i}{2})}^{3} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.41

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.1.41.1

Apply the product rule to $- \frac{\sqrt{2} i}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} ({(- 1)}^{3} {(\frac{\sqrt{2} i}{2})}^{3}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.41.2

Apply the product rule to $\frac{\sqrt{2} i}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} ({(- 1)}^{3} \frac{{(\sqrt{2} i)}^{3}}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.41.3

Apply the product rule to $\sqrt{2} i$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} ({(- 1)}^{3} \frac{{\sqrt{2}}^{3} i^{3}}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.42

Raise $- 1$ to the power of $3$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{{\sqrt{2}}^{3} i^{3}}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43

Simplify the numerator.

Tap for more steps...

Step 6.1.43.1

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{\sqrt{2^{3}} i^{3}}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.2

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{\sqrt{8} i^{3}}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.3

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

Tap for more steps...

Step 6.1.43.3.1

Factor $4$ out of $8$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{\sqrt{4 (2)} i^{3}}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.3.2

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{\sqrt{2^{2} \cdot 2} i^{3}}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.4

Pull terms out from under the radical.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{2 \sqrt{2} i^{3}}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.5

Factor out $i^{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{2 \sqrt{2} (i^{2} \cdot i)}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.6

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{2 \sqrt{2} (- 1 \cdot i)}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.7

Rewrite $- 1 i$ as $- i$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{2 \sqrt{2} \cdot - 1 i}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.8

Combine exponents.

Tap for more steps...

Step 6.1.43.8.1

Factor out negative.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{- (2 \sqrt{2} i)}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.43.8.2

Multiply $2$ by $- 1$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{- 2 (\sqrt{2} i)}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{- 2 \sqrt{2} i}{2^{3}}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.44

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} (- \frac{- 2 \sqrt{2} i}{8}) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.45

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.45.1

Move the leading negative in $- \frac{- 2 \sqrt{2} i}{8}$ into the numerator.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - 2 \sqrt{6} \frac{2 \sqrt{2} i}{8} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.45.2

Factor $2$ out of $- 2 \sqrt{6}$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} + 2 (- \sqrt{6}) \frac{2 \sqrt{2} i}{8} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.45.3

Factor $2$ out of $8$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} + 2 (- \sqrt{6}) \frac{2 \sqrt{2} i}{2 (4)} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.45.4

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} + 2 (- \sqrt{6}) \frac{2 \sqrt{2} i}{2 \cdot 4} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.45.5

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{6} \frac{2 \sqrt{2} i}{4} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.46

Combine $\frac{2 \sqrt{2} i}{4}$ and $\sqrt{6}$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{2 \sqrt{2} i \sqrt{6}}{4} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.47

Combine using the product rule for radicals.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{2 \sqrt{6 \cdot 2} i}{4} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.48

Multiply $6$ by $2$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{2 \sqrt{12} i}{4} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.49

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

Tap for more steps...

Step 6.1.49.1

Factor $2$ out of $2 \sqrt{12} i$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{2 (\sqrt{12} i)}{4} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.49.2

Cancel the common factors.

Tap for more steps...

Step 6.1.49.2.1

Factor $2$ out of $4$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{2 (\sqrt{12} i)}{2 \cdot 2} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.49.2.2

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{2 (\sqrt{12} i)}{2 \cdot 2} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.49.2.3

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{\sqrt{12} i}{2} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.50

Simplify the numerator.

Tap for more steps...

Step 6.1.50.1

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 6.1.50.1.1

Factor $4$ out of $12$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{\sqrt{4 (3)} i}{2} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.50.1.2

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{\sqrt{2^{2} \cdot 3} i}{2} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.50.2

Pull terms out from under the radical.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{2 \sqrt{3} i}{2} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.51

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.51.1

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \frac{2 \sqrt{3} i}{2} + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.51.2

Divide $\sqrt{3} i$ by $1$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - (\sqrt{3} i) + {(- \frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.52

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.1.52.1

Apply the product rule to $- \frac{\sqrt{2} i}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + {(- 1)}^{4} {(\frac{\sqrt{2} i}{2})}^{4}$

Step 6.1.52.2

Apply the product rule to $\frac{\sqrt{2} i}{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + {(- 1)}^{4} \frac{{(\sqrt{2} i)}^{4}}{2^{4}}$

Step 6.1.52.3

Apply the product rule to $\sqrt{2} i$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + {(- 1)}^{4} \frac{{\sqrt{2}}^{4} i^{4}}{2^{4}}$

Step 6.1.53

Raise $- 1$ to the power of $4$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + 1 \frac{{\sqrt{2}}^{4} i^{4}}{2^{4}}$

Step 6.1.54

Multiply $\frac{{\sqrt{2}}^{4} i^{4}}{2^{4}}$ by $1$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{{\sqrt{2}}^{4} i^{4}}{2^{4}}$

Step 6.1.55

Simplify the numerator.

Tap for more steps...

Step 6.1.55.1

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

Tap for more steps...

Step 6.1.55.1.1

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{{(2^{\frac{1}{2}})}^{4} i^{4}}{2^{4}}$

Step 6.1.55.1.2

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{2^{\frac{1}{2} \cdot 4} i^{4}}{2^{4}}$

Step 6.1.55.1.3

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{2^{\frac{4}{2}} i^{4}}{2^{4}}$

Step 6.1.55.1.4

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

Tap for more steps...

Step 6.1.55.1.4.1

Factor $2$ out of $4$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{2^{\frac{2 \cdot 2}{2}} i^{4}}{2^{4}}$

Step 6.1.55.1.4.2

Cancel the common factors.

Tap for more steps...

Step 6.1.55.1.4.2.1

Factor $2$ out of $2$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{2^{\frac{2 \cdot 2}{2 (1)}} i^{4}}{2^{4}}$

Step 6.1.55.1.4.2.2

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{2^{\frac{2 \cdot 2}{2 \cdot 1}} i^{4}}{2^{4}}$

Step 6.1.55.1.4.2.3

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{2^{\frac{2}{1}} i^{4}}{2^{4}}$

Step 6.1.55.1.4.2.4

Divide $2$ by $1$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{2^{2} i^{4}}{2^{4}}$

Step 6.1.55.2

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4 i^{4}}{2^{4}}$

Step 6.1.55.3

Rewrite $i^{4}$ as $1$ .

Tap for more steps...

Step 6.1.55.3.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4 {(i^{2})}^{2}}{2^{4}}$

Step 6.1.55.3.2

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

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4 {(- 1)}^{2}}{2^{4}}$

Step 6.1.55.3.3

Raise $- 1$ to the power of $2$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4 \cdot 1}{2^{4}}$

Step 6.1.56

Raise $2$ to the power of $4$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4 \cdot 1}{16}$

Step 6.1.57

Multiply $4$ by $1$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4}{16}$

Step 6.1.58

Cancel the common factor of $4$ and $16$ .

Tap for more steps...

Step 6.1.58.1

Factor $4$ out of $4$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4 (1)}{16}$

Step 6.1.58.2

Cancel the common factors.

Tap for more steps...

Step 6.1.58.2.1

Factor $4$ out of $16$ .

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4 \cdot 1}{4 \cdot 4}$

Step 6.1.58.2.2

Cancel the common factor.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{4 \cdot 1}{4 \cdot 4}$

Step 6.1.58.2.3

Rewrite the expression.

$\frac{9}{4} + 3 \sqrt{3} i - \frac{9}{2} - \sqrt{3} i + \frac{1}{4}$

Step 6.2

Combine fractions.

Tap for more steps...

Step 6.2.1

Combine the numerators over the common denominator.

$3 \sqrt{3} i - \sqrt{3} i + \frac{9 + 1}{4} + \frac{- 9}{2}$

Step 6.2.2

Add $9$ and $1$ .

$3 \sqrt{3} i - \sqrt{3} i + \frac{10}{4} + \frac{- 9}{2}$

Step 6.3

Simplify each term.

Tap for more steps...

Step 6.3.1

Cancel the common factor of $10$ and $4$ .

Tap for more steps...

Step 6.3.1.1

Factor $2$ out of $10$ .

$3 \sqrt{3} i - \sqrt{3} i + \frac{2 (5)}{4} + \frac{- 9}{2}$

Step 6.3.1.2

Cancel the common factors.

Tap for more steps...

Step 6.3.1.2.1

Factor $2$ out of $4$ .

$3 \sqrt{3} i - \sqrt{3} i + \frac{2 \cdot 5}{2 \cdot 2} + \frac{- 9}{2}$

Step 6.3.1.2.2

Cancel the common factor.

$3 \sqrt{3} i - \sqrt{3} i + \frac{2 \cdot 5}{2 \cdot 2} + \frac{- 9}{2}$

Step 6.3.1.2.3

Rewrite the expression.

$3 \sqrt{3} i - \sqrt{3} i + \frac{5}{2} + \frac{- 9}{2}$

Step 6.3.2

Move the negative in front of the fraction.

$3 \sqrt{3} i - \sqrt{3} i + \frac{5}{2} - \frac{9}{2}$

Step 6.4

Simplify terms.

Tap for more steps...

Step 6.4.1

Combine the numerators over the common denominator.

$3 \sqrt{3} i - \sqrt{3} i + \frac{5 - 9}{2}$

Step 6.4.2

Simplify the expression.

Tap for more steps...

Step 6.4.2.1

Subtract $9$ from $5$ .

$3 \sqrt{3} i - \sqrt{3} i + \frac{- 4}{2}$

Step 6.4.2.2

Divide $- 4$ by $2$ .

$3 \sqrt{3} i - \sqrt{3} i - 2$

Step 6.4.3

Subtract $\sqrt{3} i$ from $3 \sqrt{3} i$ .

$2 \sqrt{3} i - 2$

Step 6.4.4

Reorder $2 \sqrt{3} i$ and $- 2$ .

$- 2 + 2 \sqrt{3} i$