Precalculus Examples

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

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.

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

Step 1.2

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

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

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.

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

Apply the distributive property.

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 3.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

Rewrite the expression.

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

Step 4.2.5

Evaluate the exponent.

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

Step 5

Use the Binomial Theorem.

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

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 $- \frac{3}{2}$ .

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

Step 6.1.1.2

Apply the product rule to $\frac{3}{2}$ .

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

Step 6.1.2

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

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

Step 6.1.3

Multiply $\frac{3^{4}}{2^{4}}$ by $1$ .

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

Step 6.1.4

Raise $3$ to the power of $4$ .

$\frac{81}{2^{4}} + 4 {(- \frac{3}{2})}^{3} (- \frac{\sqrt{3} i}{2}) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.5

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

$\frac{81}{16} + 4 {(- \frac{3}{2})}^{3} (- \frac{\sqrt{3} i}{2}) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.6.1

Move the leading negative in $- \frac{\sqrt{3} i}{2}$ into the numerator.

$\frac{81}{16} + 4 {(- \frac{3}{2})}^{3} \frac{- \sqrt{3} i}{2} + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.6.2

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

$\frac{81}{16} + 2 (2 {(- \frac{3}{2})}^{3}) \frac{- \sqrt{3} i}{2} + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.6.3

Cancel the common factor.

Step 6.1.6.4

Rewrite the expression.

$\frac{81}{16} + 2 {(- \frac{3}{2})}^{3} (- \sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.7

Multiply $- 1$ by $2$ .

$\frac{81}{16} - 2 {(- \frac{3}{2})}^{3} (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.8

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

Tap for more steps...

Step 6.1.8.1

Apply the product rule to $- \frac{3}{2}$ .

$\frac{81}{16} - 2 ({(- 1)}^{3} {(\frac{3}{2})}^{3}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.8.2

Apply the product rule to $\frac{3}{2}$ .

$\frac{81}{16} - 2 ({(- 1)}^{3} \frac{3^{3}}{2^{3}}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.9

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

$\frac{81}{16} - 2 (- \frac{3^{3}}{2^{3}}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.10

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

$\frac{81}{16} - 2 (- \frac{27}{2^{3}}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.11

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

$\frac{81}{16} - 2 (- \frac{27}{8}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.12

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.12.1

Move the leading negative in $- \frac{27}{8}$ into the numerator.

$\frac{81}{16} - 2 (\frac{- 27}{8}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.12.2

Factor $2$ out of $- 2$ .

$\frac{81}{16} + 2 (- 1) \frac{- 27}{8} (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.12.3

Factor $2$ out of $8$ .

$\frac{81}{16} + 2 \cdot - 1 \frac{- 27}{2 \cdot 4} (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.12.4

Cancel the common factor.

Step 6.1.12.5

Rewrite the expression.

$\frac{81}{16} - 1 (\frac{- 27}{4}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.13

Move the negative in front of the fraction.

$\frac{81}{16} - 1 (- \frac{27}{4}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.14

Multiply $- 1 (- \frac{27}{4})$ .

Tap for more steps...

Step 6.1.14.1

Multiply $- 1$ by $- 1$ .

$\frac{81}{16} + 1 (\frac{27}{4}) (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.14.2

Multiply $\frac{27}{4}$ by $1$ .

$\frac{81}{16} + \frac{27}{4} (\sqrt{3} i) + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.15

Multiply $\frac{27}{4} (\sqrt{3} i)$ .

Tap for more steps...

Step 6.1.15.1

Combine $\sqrt{3}$ and $\frac{27}{4}$ .

$\frac{81}{16} + \frac{\sqrt{3} \cdot 27}{4} i + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.15.2

Combine $\frac{\sqrt{3} \cdot 27}{4}$ and $i$ .

$\frac{81}{16} + \frac{\sqrt{3} \cdot 27 i}{4} + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.16

Move $27$ to the left of $\sqrt{3}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 6 {(- \frac{3}{2})}^{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.17

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

Tap for more steps...

Step 6.1.17.1

Apply the product rule to $- \frac{3}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 6 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.17.2

Apply the product rule to $\frac{3}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 6 ({(- 1)}^{2} \frac{3^{2}}{2^{2}}) {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.18

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 6 (1 \frac{3^{2}}{2^{2}}) {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.19

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 6 \frac{3^{2}}{2^{2}} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.20

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 6 \frac{9}{2^{2}} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.21

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 6 (\frac{9}{4}) {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.22

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.22.1

Factor $2$ out of $6$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 2 (3) \frac{9}{4} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.22.2

Factor $2$ out of $4$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 2 \cdot 3 \frac{9}{2 \cdot 2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.22.3

Cancel the common factor.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 2 \cdot 3 \frac{9}{2 \cdot 2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.22.4

Rewrite the expression.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + 3 (\frac{9}{2}) {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.23

Combine $3$ and $\frac{9}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{3 \cdot 9}{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.24

Multiply $3$ by $9$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27}{2} {(- \frac{\sqrt{3} i}{2})}^{2} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.25

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

Tap for more steps...

Step 6.1.25.1

Apply the product rule to $- \frac{\sqrt{3} i}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27}{2} ({(- 1)}^{2} {(\frac{\sqrt{3} i}{2})}^{2}) + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.25.2

Apply the product rule to $\frac{\sqrt{3} i}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27}{2} ({(- 1)}^{2} \frac{{(\sqrt{3} i)}^{2}}{2^{2}}) + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.25.3

Apply the product rule to $\sqrt{3} i$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27}{2} ({(- 1)}^{2} \frac{{\sqrt{3}}^{2} i^{2}}{2^{2}}) + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.26

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27}{2} (1 \frac{{\sqrt{3}}^{2} i^{2}}{2^{2}}) + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.27

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27}{2} \cdot \frac{{\sqrt{3}}^{2} i^{2}}{2^{2}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.28

Combine.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 ({\sqrt{3}}^{2} i^{2})}{2 \cdot 2^{2}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.29

Multiply $2$ by $2^{2}$ by adding the exponents.

Tap for more steps...

Step 6.1.29.1

Multiply $2$ by $2^{2}$ .

Tap for more steps...

Step 6.1.29.1.1

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 ({\sqrt{3}}^{2} i^{2})}{2^{1} \cdot 2^{2}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.29.1.2

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 ({\sqrt{3}}^{2} i^{2})}{2^{1 + 2}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.29.2

Add $1$ and $2$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 ({\sqrt{3}}^{2} i^{2})}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30

Simplify the numerator.

Tap for more steps...

Step 6.1.30.1

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

Tap for more steps...

Step 6.1.30.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 {(3^{\frac{1}{2}})}^{2} i^{2}}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30.1.2

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 \cdot 3^{\frac{1}{2} \cdot 2} i^{2}}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30.1.3

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 \cdot 3^{\frac{2}{2}} i^{2}}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.30.1.4.1

Cancel the common factor.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 \cdot 3^{\frac{2}{2}} i^{2}}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30.1.4.2

Rewrite the expression.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 \cdot 3^{1} i^{2}}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30.1.5

Evaluate the exponent.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 \cdot 3 i^{2}}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30.2

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{27 \cdot 3 \cdot - 1}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30.3

Combine exponents.

Tap for more steps...

Step 6.1.30.3.1

Multiply $27$ by $3$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{81 \cdot - 1}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.30.3.2

Multiply $81$ by $- 1$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{- 81}{2^{3}} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.31

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} + \frac{- 81}{8} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.32

Move the negative in front of the fraction.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + 4 (- \frac{3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.33

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.33.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + 4 (\frac{- 3}{2}) {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.33.2

Factor $2$ out of $4$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + 2 (2) \frac{- 3}{2} {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.33.3

Cancel the common factor.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + 2 \cdot 2 \frac{- 3}{2} {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.33.4

Rewrite the expression.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + 2 \cdot - 3 {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.34

Multiply $2$ by $- 3$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 {(- \frac{\sqrt{3} i}{2})}^{3} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.35

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

Tap for more steps...

Step 6.1.35.1

Apply the product rule to $- \frac{\sqrt{3} i}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 ({(- 1)}^{3} {(\frac{\sqrt{3} i}{2})}^{3}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.35.2

Apply the product rule to $\frac{\sqrt{3} i}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 ({(- 1)}^{3} \frac{{(\sqrt{3} i)}^{3}}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.35.3

Apply the product rule to $\sqrt{3} i$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 ({(- 1)}^{3} \frac{{\sqrt{3}}^{3} i^{3}}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.36

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{{\sqrt{3}}^{3} i^{3}}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37

Simplify the numerator.

Tap for more steps...

Step 6.1.37.1

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{\sqrt{3^{3}} i^{3}}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.2

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{\sqrt{27} i^{3}}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.3

Rewrite $27$ as $3^{2} \cdot 3$ .

Tap for more steps...

Step 6.1.37.3.1

Factor $9$ out of $27$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{\sqrt{9 (3)} i^{3}}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.3.2

Rewrite $9$ as $3^{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{\sqrt{3^{2} \cdot 3} i^{3}}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.4

Pull terms out from under the radical.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{3 \sqrt{3} i^{3}}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.5

Factor out $i^{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{3 \sqrt{3} (i^{2} \cdot i)}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.6

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{3 \sqrt{3} (- 1 \cdot i)}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.7

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{3 \sqrt{3} \cdot - 1 i}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.8

Combine exponents.

Tap for more steps...

Step 6.1.37.8.1

Factor out negative.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{- (3 \sqrt{3} i)}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.37.8.2

Multiply $3$ by $- 1$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{- 3 (\sqrt{3} i)}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{- 3 \sqrt{3} i}{2^{3}}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.38

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 (- \frac{- 3 \sqrt{3} i}{8}) + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.39

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.39.1

Move the leading negative in $- \frac{- 3 \sqrt{3} i}{8}$ into the numerator.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 6 \frac{3 \sqrt{3} i}{8} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.39.2

Factor $2$ out of $- 6$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + 2 (- 3) \frac{3 \sqrt{3} i}{8} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.39.3

Factor $2$ out of $8$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + 2 \cdot - 3 \frac{3 \sqrt{3} i}{2 \cdot 4} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.39.4

Cancel the common factor.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + 2 \cdot - 3 \frac{3 \sqrt{3} i}{2 \cdot 4} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.39.5

Rewrite the expression.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - 3 \frac{3 \sqrt{3} i}{4} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.40

Combine $- 3$ and $\frac{3 \sqrt{3} i}{4}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + \frac{- 3 (3 \sqrt{3} i)}{4} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.41

Multiply $3$ by $- 3$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} + \frac{- 9 (\sqrt{3} i)}{4} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.42

Move the negative in front of the fraction.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{(9) \sqrt{3} i}{4} + {(- \frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.43

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

Tap for more steps...

Step 6.1.43.1

Apply the product rule to $- \frac{\sqrt{3} i}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + {(- 1)}^{4} {(\frac{\sqrt{3} i}{2})}^{4}$

Step 6.1.43.2

Apply the product rule to $\frac{\sqrt{3} i}{2}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + {(- 1)}^{4} \frac{{(\sqrt{3} i)}^{4}}{2^{4}}$

Step 6.1.43.3

Apply the product rule to $\sqrt{3} i$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + {(- 1)}^{4} \frac{{\sqrt{3}}^{4} i^{4}}{2^{4}}$

Step 6.1.44

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + 1 \frac{{\sqrt{3}}^{4} i^{4}}{2^{4}}$

Step 6.1.45

Multiply $\frac{{\sqrt{3}}^{4} i^{4}}{2^{4}}$ by $1$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{{\sqrt{3}}^{4} i^{4}}{2^{4}}$

Step 6.1.46

Simplify the numerator.

Tap for more steps...

Step 6.1.46.1

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

Tap for more steps...

Step 6.1.46.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{{(3^{\frac{1}{2}})}^{4} i^{4}}{2^{4}}$

Step 6.1.46.1.2

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{3^{\frac{1}{2} \cdot 4} i^{4}}{2^{4}}$

Step 6.1.46.1.3

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{3^{\frac{4}{2}} i^{4}}{2^{4}}$

Step 6.1.46.1.4

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

Tap for more steps...

Step 6.1.46.1.4.1

Factor $2$ out of $4$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{3^{\frac{2 \cdot 2}{2}} i^{4}}{2^{4}}$

Step 6.1.46.1.4.2

Cancel the common factors.

Tap for more steps...

Step 6.1.46.1.4.2.1

Factor $2$ out of $2$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{3^{\frac{2 \cdot 2}{2 (1)}} i^{4}}{2^{4}}$

Step 6.1.46.1.4.2.2

Cancel the common factor.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{3^{\frac{2 \cdot 2}{2 \cdot 1}} i^{4}}{2^{4}}$

Step 6.1.46.1.4.2.3

Rewrite the expression.

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{3^{\frac{2}{1}} i^{4}}{2^{4}}$

Step 6.1.46.1.4.2.4

Divide $2$ by $1$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{3^{2} i^{4}}{2^{4}}$

Step 6.1.46.2

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{9 i^{4}}{2^{4}}$

Step 6.1.46.3

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

Tap for more steps...

Step 6.1.46.3.1

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{9 {(i^{2})}^{2}}{2^{4}}$

Step 6.1.46.3.2

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{9 {(- 1)}^{2}}{2^{4}}$

Step 6.1.46.3.3

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{9 \cdot 1}{2^{4}}$

Step 6.1.47

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

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{9 \cdot 1}{16}$

Step 6.1.48

Multiply $9$ by $1$ .

$\frac{81}{16} + \frac{27 \sqrt{3} i}{4} - \frac{81}{8} - \frac{9 \sqrt{3} i}{4} + \frac{9}{16}$

Step 6.2

Simplify terms.

Tap for more steps...

Step 6.2.1

Combine the numerators over the common denominator.

$\frac{81 + 9}{16} + \frac{27 \sqrt{3} i - 9 \sqrt{3} i}{4} + \frac{- 81}{8}$

Step 6.2.2

Add $81$ and $9$ .

$\frac{90}{16} + \frac{27 \sqrt{3} i - 9 \sqrt{3} i}{4} + \frac{- 81}{8}$

Step 6.2.3

Subtract $9 \sqrt{3} i$ from $27 \sqrt{3} i$ .

$\frac{90}{16} + \frac{18 \sqrt{3} i}{4} + \frac{- 81}{8}$

Step 6.3

Simplify each term.

Tap for more steps...

Step 6.3.1

Cancel the common factor of $90$ and $16$ .

Tap for more steps...

Step 6.3.1.1

Factor $2$ out of $90$ .

$\frac{2 (45)}{16} + \frac{18 \sqrt{3} i}{4} + \frac{- 81}{8}$

Step 6.3.1.2

Cancel the common factors.

Tap for more steps...

Step 6.3.1.2.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 45}{2 \cdot 8} + \frac{18 \sqrt{3} i}{4} + \frac{- 81}{8}$

Step 6.3.1.2.2

Cancel the common factor.

$\frac{2 \cdot 45}{2 \cdot 8} + \frac{18 \sqrt{3} i}{4} + \frac{- 81}{8}$

Step 6.3.1.2.3

Rewrite the expression.

$\frac{45}{8} + \frac{18 \sqrt{3} i}{4} + \frac{- 81}{8}$

Step 6.3.2

Cancel the common factor of $18$ and $4$ .

Tap for more steps...

Step 6.3.2.1

Factor $2$ out of $18 \sqrt{3} i$ .

$\frac{45}{8} + \frac{2 (9 \sqrt{3} i)}{4} + \frac{- 81}{8}$

Step 6.3.2.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.2.1

Factor $2$ out of $4$ .

$\frac{45}{8} + \frac{2 (9 \sqrt{3} i)}{2 (2)} + \frac{- 81}{8}$

Step 6.3.2.2.2

Cancel the common factor.

$\frac{45}{8} + \frac{2 (9 \sqrt{3} i)}{2 \cdot 2} + \frac{- 81}{8}$

Step 6.3.2.2.3

Rewrite the expression.

$\frac{45}{8} + \frac{9 \sqrt{3} i}{2} + \frac{- 81}{8}$

Step 6.3.3

Move the negative in front of the fraction.

$\frac{45}{8} + \frac{9 \sqrt{3} i}{2} - \frac{81}{8}$

Step 6.4

Combine fractions.

Tap for more steps...

Step 6.4.1

Combine the numerators over the common denominator.

$\frac{45 - 81}{8} + \frac{9 \sqrt{3} i}{2}$

Step 6.4.2

Subtract $81$ from $45$ .

$\frac{- 36}{8} + \frac{9 \sqrt{3} i}{2}$

Step 6.5

Simplify each term.

Tap for more steps...

Step 6.5.1

Cancel the common factor of $- 36$ and $8$ .

Tap for more steps...

Step 6.5.1.1

Factor $4$ out of $- 36$ .

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

Step 6.5.1.2

Cancel the common factors.

Tap for more steps...

Step 6.5.1.2.1

Factor $4$ out of $8$ .

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

Step 6.5.1.2.2

Cancel the common factor.

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

Step 6.5.1.2.3

Rewrite the expression.

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

Step 6.5.2

Move the negative in front of the fraction.

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

Step 7

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 8

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 9

Substitute the actual values of $a = - \frac{9}{2}$ and $b = \frac{9 \sqrt{3}}{2}$ .

$| z | = \sqrt{{(\frac{9 \sqrt{3}}{2})}^{2} + {(- \frac{9}{2})}^{2}}$

Step 10

Find $| z |$ .

Tap for more steps...

Step 10.1

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

Tap for more steps...

Step 10.1.1

Apply the product rule to $\frac{9 \sqrt{3}}{2}$ .

$| z | = \sqrt{\frac{{(9 \sqrt{3})}^{2}}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.1.2

Apply the product rule to $9 \sqrt{3}$ .

$| z | = \sqrt{\frac{9^{2} {\sqrt{3}}^{2}}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.2

Simplify the numerator.

Tap for more steps...

Step 10.2.1

Raise $9$ to the power of $2$ .

$| z | = \sqrt{\frac{81 {\sqrt{3}}^{2}}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.2.2

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

Tap for more steps...

Step 10.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$| z | = \sqrt{\frac{81 {(3^{\frac{1}{2}})}^{2}}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.2.2.2

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

$| z | = \sqrt{\frac{81 \cdot 3^{\frac{1}{2} \cdot 2}}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.2.2.3

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

$| z | = \sqrt{\frac{81 \cdot 3^{\frac{2}{2}}}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.2.4.1

Cancel the common factor.

$| z | = \sqrt{\frac{81 \cdot 3^{\frac{2}{2}}}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.2.2.4.2

Rewrite the expression.

$| z | = \sqrt{\frac{81 \cdot 3}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.2.2.5

Evaluate the exponent.

$| z | = \sqrt{\frac{81 \cdot 3}{2^{2}} + {(- \frac{9}{2})}^{2}}$

Step 10.3

Simplify the expression.

Tap for more steps...

Step 10.3.1

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

$| z | = \sqrt{\frac{81 \cdot 3}{4} + {(- \frac{9}{2})}^{2}}$

Step 10.3.2

Multiply $81$ by $3$ .

$| z | = \sqrt{\frac{243}{4} + {(- \frac{9}{2})}^{2}}$

Step 10.4

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

Tap for more steps...

Step 10.4.1

Apply the product rule to $- \frac{9}{2}$ .

$| z | = \sqrt{\frac{243}{4} + {(- 1)}^{2} {(\frac{9}{2})}^{2}}$

Step 10.4.2

Apply the product rule to $\frac{9}{2}$ .

$| z | = \sqrt{\frac{243}{4} + {(- 1)}^{2} (\frac{9^{2}}{2^{2}})}$

Step 10.5

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

$| z | = \sqrt{\frac{243}{4} + 1 (\frac{9^{2}}{2^{2}})}$

Step 10.6

Multiply $\frac{9^{2}}{2^{2}}$ by $1$ .

$| z | = \sqrt{\frac{243}{4} + \frac{9^{2}}{2^{2}}}$

Step 10.7

Raise $9$ to the power of $2$ .

$| z | = \sqrt{\frac{243}{4} + \frac{81}{2^{2}}}$

Step 10.8

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

$| z | = \sqrt{\frac{243}{4} + \frac{81}{4}}$

Step 10.9

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{243 + 81}{4}}$

Step 10.10

Add $243$ and $81$ .

$| z | = \sqrt{\frac{324}{4}}$

Step 10.11

Divide $324$ by $4$ .

$| z | = \sqrt{81}$

Step 10.12

Rewrite $81$ as $9^{2}$ .

$| z | = \sqrt{9^{2}}$

Step 10.13

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 9$

Step 11

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{\frac{9 \sqrt{3}}{2}}{- \frac{9}{2}})$

Step 12

Since inverse tangent of $\frac{\frac{9 \sqrt{3}}{2}}{- \frac{9}{2}}$ produces an angle in the second quadrant, the value of the angle is $\frac{2 π}{3}$ .

$θ = \frac{2 π}{3}$

Step 13

Substitute the values of $θ = \frac{2 π}{3}$ and $| z | = 9$ .

$9 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3}))$