Precalculus Examples

${(\sqrt{3} + i)}^{5}$

Step 1

Use the Binomial Theorem.

${\sqrt{3}}^{5} + 5 {\sqrt{3}}^{4} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${\sqrt{3}}^{5}$ as $\sqrt{3^{5}}$ .

$\sqrt{3^{5}} + 5 {\sqrt{3}}^{4} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.2

Raise $3$ to the power of $5$ .

$\sqrt{243} + 5 {\sqrt{3}}^{4} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.3

Rewrite $243$ as $9^{2} \cdot 3$ .

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Step 2.1.3.1

Factor $81$ out of $243$ .

$\sqrt{81 (3)} + 5 {\sqrt{3}}^{4} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.3.2

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

$\sqrt{9^{2} \cdot 3} + 5 {\sqrt{3}}^{4} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.4

Pull terms out from under the radical.

$9 \sqrt{3} + 5 {\sqrt{3}}^{4} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.5

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

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Step 2.1.5.1

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

$9 \sqrt{3} + 5 {(3^{\frac{1}{2}})}^{4} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.5.2

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

$9 \sqrt{3} + 5 \cdot 3^{\frac{1}{2} \cdot 4} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.5.3

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

$9 \sqrt{3} + 5 \cdot 3^{\frac{4}{2}} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.5.4

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

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Step 2.1.5.4.1

Factor $2$ out of $4$ .

$9 \sqrt{3} + 5 \cdot 3^{\frac{2 \cdot 2}{2}} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.5.4.2

Cancel the common factors.

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Step 2.1.5.4.2.1

Factor $2$ out of $2$ .

$9 \sqrt{3} + 5 \cdot 3^{\frac{2 \cdot 2}{2 (1)}} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.5.4.2.2

Cancel the common factor.

$9 \sqrt{3} + 5 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.5.4.2.3

Rewrite the expression.

$9 \sqrt{3} + 5 \cdot 3^{\frac{2}{1}} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.5.4.2.4

Divide $2$ by $1$ .

$9 \sqrt{3} + 5 \cdot 3^{2} i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.6

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

$9 \sqrt{3} + 5 \cdot 9 i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.7

Multiply $5$ by $9$ .

$9 \sqrt{3} + 45 i + 10 {\sqrt{3}}^{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.8

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

$9 \sqrt{3} + 45 i + 10 \sqrt{3^{3}} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.9

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

$9 \sqrt{3} + 45 i + 10 \sqrt{27} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.10

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

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Step 2.1.10.1

Factor $9$ out of $27$ .

$9 \sqrt{3} + 45 i + 10 \sqrt{9 (3)} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.10.2

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

$9 \sqrt{3} + 45 i + 10 \sqrt{3^{2} \cdot 3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.11

Pull terms out from under the radical.

$9 \sqrt{3} + 45 i + 10 (3 \sqrt{3}) i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.12

Multiply $3$ by $10$ .

$9 \sqrt{3} + 45 i + 30 \sqrt{3} i^{2} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.13

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

$9 \sqrt{3} + 45 i + 30 \sqrt{3} \cdot - 1 + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.14

Multiply $- 1$ by $30$ .

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 10 {\sqrt{3}}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.15

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

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Step 2.1.15.1

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 10 {(3^{\frac{1}{2}})}^{2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.15.2

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 10 \cdot 3^{\frac{1}{2} \cdot 2} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.15.3

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 10 \cdot 3^{\frac{2}{2}} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.15.4

Cancel the common factor of $2$ .

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Step 2.1.15.4.1

Cancel the common factor.

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 10 \cdot 3^{\frac{2}{2}} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.15.4.2

Rewrite the expression.

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 10 \cdot 3^{1} i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.15.5

Evaluate the exponent.

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 10 \cdot 3 i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.16

Multiply $10$ by $3$ .

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 30 i^{3} + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.17

Factor out $i^{2}$ .

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 30 (i^{2} \cdot i) + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.18

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 30 (- 1 \cdot i) + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.19

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} + 30 (- i) + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.20

Multiply $- 1$ by $30$ .

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} i^{4} + i^{5}$

Step 2.1.21

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

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Step 2.1.21.1

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} {(i^{2})}^{2} + i^{5}$

Step 2.1.21.2

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} {(- 1)}^{2} + i^{5}$

Step 2.1.21.3

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} \cdot 1 + i^{5}$

Step 2.1.22

Multiply $5$ by $1$ .

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} + i^{5}$

Step 2.1.23

Factor out $i^{4}$ .

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} + i^{4} i$

Step 2.1.24

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

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Step 2.1.24.1

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} + {(i^{2})}^{2} i$

Step 2.1.24.2

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} + {(- 1)}^{2} i$

Step 2.1.24.3

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

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} + 1 i$

Step 2.1.25

Multiply $i$ by $1$ .

$9 \sqrt{3} + 45 i - 30 \sqrt{3} - 30 i + 5 \sqrt{3} + i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $30 \sqrt{3}$ from $9 \sqrt{3}$ .

$45 i - 21 \sqrt{3} - 30 i + 5 \sqrt{3} + i$

Step 2.2.2

Subtract $30 i$ from $45 i$ .

$15 i - 21 \sqrt{3} + 5 \sqrt{3} + i$

Step 2.2.3

Add $15 i$ and $i$ .

$16 i - 21 \sqrt{3} + 5 \sqrt{3}$

Step 2.2.4

Add $- 21 \sqrt{3}$ and $5 \sqrt{3}$ .

$16 i - 16 \sqrt{3}$

Step 2.2.5

Reorder $16 i$ and $- 16 \sqrt{3}$ .

$- 16 \sqrt{3} + 16 i$

Step 3

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 4

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 5

Substitute the actual values of $a = - 16 \sqrt{3}$ and $b = 16$ .

$| z | = \sqrt{16^{2} + {(- 16 \sqrt{3})}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Simplify the expression.

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Step 6.1.1

Raise $16$ to the power of $2$ .

$| z | = \sqrt{256 + {(- 16 \sqrt{3})}^{2}}$

Step 6.1.2

Apply the product rule to $- 16 \sqrt{3}$ .

$| z | = \sqrt{256 + {(- 16)}^{2} {\sqrt{3}}^{2}}$

Step 6.1.3

Raise $- 16$ to the power of $2$ .

$| z | = \sqrt{256 + 256 {\sqrt{3}}^{2}}$

Step 6.2

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

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Step 6.2.1

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

$| z | = \sqrt{256 + 256 {(3^{\frac{1}{2}})}^{2}}$

Step 6.2.2

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

$| z | = \sqrt{256 + 256 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 6.2.3

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

$| z | = \sqrt{256 + 256 \cdot 3^{\frac{2}{2}}}$

Step 6.2.4

Cancel the common factor of $2$ .

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Step 6.2.4.1

Cancel the common factor.

$| z | = \sqrt{256 + 256 \cdot 3^{\frac{2}{2}}}$

Step 6.2.4.2

Rewrite the expression.

$| z | = \sqrt{256 + 256 \cdot 3}$

Step 6.2.5

Evaluate the exponent.

$| z | = \sqrt{256 + 256 \cdot 3}$

Step 6.3

Simplify the expression.

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Step 6.3.1

Multiply $256$ by $3$ .

$| z | = \sqrt{256 + 768}$

Step 6.3.2

Add $256$ and $768$ .

$| z | = \sqrt{1024}$

Step 6.3.3

Rewrite $1024$ as $32^{2}$ .

$| z | = \sqrt{32^{2}}$

Step 6.3.4

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

$| z | = 32$

Step 7

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

$θ = \arctan (\frac{16}{- 16 \sqrt{3}})$

Step 8

Since inverse tangent of $\frac{16}{- 16 \sqrt{3}}$ produces an angle in the second quadrant, the value of the angle is $\frac{5 π}{6}$ .

$θ = \frac{5 π}{6}$

Step 9

Substitute the values of $θ = \frac{5 π}{6}$ and $| z | = 32$ .

$32 (\cos (\frac{5 π}{6}) + i \sin (\frac{5 π}{6}))$