Trigonometry Examples

$\frac{7 \sqrt{21} + 7 i \sqrt{7}}{\sqrt{7} + i \sqrt{21}}$

Step 1

Multiply $\frac{7 \sqrt{21} + 7 i \sqrt{7}}{\sqrt{7} + i \sqrt{21}}$ by $\frac{\sqrt{7} - i \sqrt{21}}{\sqrt{7} - i \sqrt{21}}$ .

$\frac{7 \sqrt{21} + 7 i \sqrt{7}}{\sqrt{7} + i \sqrt{21}} \cdot \frac{\sqrt{7} - i \sqrt{21}}{\sqrt{7} - i \sqrt{21}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{7 \sqrt{21} + 7 i \sqrt{7}}{\sqrt{7} + i \sqrt{21}}$ by $\frac{\sqrt{7} - i \sqrt{21}}{\sqrt{7} - i \sqrt{21}}$ .

$\frac{(7 \sqrt{21} + 7 i \sqrt{7}) (\sqrt{7} - i \sqrt{21})}{(\sqrt{7} + i \sqrt{21}) (\sqrt{7} - i \sqrt{21})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(7 \sqrt{21} + 7 i \sqrt{7}) (\sqrt{7} - i \sqrt{21})}{{\sqrt{7}}^{2} - i \sqrt{147} + i \sqrt{147} - i^{2} {\sqrt{21}}^{2}}$

Step 2.3

Simplify.

$\frac{(7 \sqrt{21} + 7 i \sqrt{7}) (\sqrt{7} - i \sqrt{21})}{28}$

Step 2.4

Cancel the common factor of $7 \sqrt{21} + 7 i \sqrt{7}$ and $28$ .

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Step 2.4.1

Factor $7$ out of $(7 \sqrt{21} + 7 i \sqrt{7}) (\sqrt{7} - i \sqrt{21})$ .

$\frac{7 ((\sqrt{21} + i \sqrt{7}) (\sqrt{7} - i \sqrt{21}))}{28}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $7$ out of $28$ .

$\frac{7 ((\sqrt{21} + i \sqrt{7}) (\sqrt{7} - i \sqrt{21}))}{7 (4)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{7 ((\sqrt{21} + i \sqrt{7}) (\sqrt{7} - i \sqrt{21}))}{7 \cdot 4}$

Step 2.4.2.3

Rewrite the expression.

$\frac{(\sqrt{21} + i \sqrt{7}) (\sqrt{7} - i \sqrt{21})}{4}$

Step 3

Expand $(\sqrt{21} + i \sqrt{7}) (\sqrt{7} - i \sqrt{21})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{21} (\sqrt{7} - i \sqrt{21}) + i \sqrt{7} (\sqrt{7} - i \sqrt{21})}{4}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{21} \sqrt{7} + \sqrt{21} (- i \sqrt{21}) + i \sqrt{7} (\sqrt{7} - i \sqrt{21})}{4}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{21} \sqrt{7} + \sqrt{21} (- i \sqrt{21}) + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{21 \cdot 7} + \sqrt{21} (- i \sqrt{21}) + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.2

Multiply $21$ by $7$ .

$\frac{\sqrt{147} + \sqrt{21} (- i \sqrt{21}) + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.3

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 4.1.3.1

Factor $49$ out of $147$ .

$\frac{\sqrt{49 (3)} + \sqrt{21} (- i \sqrt{21}) + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.3.2

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{7^{2} \cdot 3} + \sqrt{21} (- i \sqrt{21}) + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.4

Pull terms out from under the radical.

$\frac{7 \sqrt{3} + \sqrt{21} (- i \sqrt{21}) + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.5

Multiply $\sqrt{21} (- i \sqrt{21})$ .

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Step 4.1.5.1

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{7 \sqrt{3} - i ({\sqrt{21}}^{1} \sqrt{21}) + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.5.2

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{7 \sqrt{3} - i ({\sqrt{21}}^{1} {\sqrt{21}}^{1}) + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.5.3

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

$\frac{7 \sqrt{3} - i {\sqrt{21}}^{1 + 1} + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.5.4

Add $1$ and $1$ .

$\frac{7 \sqrt{3} - i {\sqrt{21}}^{2} + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 4.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{7 \sqrt{3} - i {(21^{\frac{1}{2}})}^{2} + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.6.2

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

$\frac{7 \sqrt{3} - i \cdot 21^{\frac{1}{2} \cdot 2} + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.6.3

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

$\frac{7 \sqrt{3} - i \cdot 21^{\frac{2}{2}} + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

$\frac{7 \sqrt{3} - i \cdot 21^{\frac{2}{2}} + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.6.4.2

Rewrite the expression.

$\frac{7 \sqrt{3} - i \cdot 21^{1} + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.6.5

Evaluate the exponent.

$\frac{7 \sqrt{3} - i \cdot 21 + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.7

Multiply $21$ by $- 1$ .

$\frac{7 \sqrt{3} - 21 i + i \sqrt{7} \sqrt{7} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.8

Multiply $i \sqrt{7} \sqrt{7}$ .

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Step 4.1.8.1

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{7 \sqrt{3} - 21 i + i ({\sqrt{7}}^{1} \sqrt{7}) + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.8.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{7 \sqrt{3} - 21 i + i ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.8.3

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

$\frac{7 \sqrt{3} - 21 i + i {\sqrt{7}}^{1 + 1} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.8.4

Add $1$ and $1$ .

$\frac{7 \sqrt{3} - 21 i + i {\sqrt{7}}^{2} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.9

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{7 \sqrt{3} - 21 i + i {(7^{\frac{1}{2}})}^{2} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.9.2

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

$\frac{7 \sqrt{3} - 21 i + i \cdot 7^{\frac{1}{2} \cdot 2} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.9.3

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

$\frac{7 \sqrt{3} - 21 i + i \cdot 7^{\frac{2}{2}} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.9.4

Cancel the common factor of $2$ .

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Step 4.1.9.4.1

Cancel the common factor.

$\frac{7 \sqrt{3} - 21 i + i \cdot 7^{\frac{2}{2}} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.9.4.2

Rewrite the expression.

$\frac{7 \sqrt{3} - 21 i + i \cdot 7^{1} + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.9.5

Evaluate the exponent.

$\frac{7 \sqrt{3} - 21 i + i \cdot 7 + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.10

Move $7$ to the left of $i$ .

$\frac{7 \sqrt{3} - 21 i + 7 \cdot i + i \sqrt{7} (- i \sqrt{21})}{4}$

Step 4.1.11

Multiply $i \sqrt{7} (- i \sqrt{21})$ .

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Step 4.1.11.1

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

$\frac{7 \sqrt{3} - 21 i + 7 i + \sqrt{7} (- (i^{1} i) \sqrt{21})}{4}$

Step 4.1.11.2

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

$\frac{7 \sqrt{3} - 21 i + 7 i + \sqrt{7} (- (i^{1} i^{1}) \sqrt{21})}{4}$

Step 4.1.11.3

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

$\frac{7 \sqrt{3} - 21 i + 7 i + \sqrt{7} (- i^{1 + 1} \sqrt{21})}{4}$

Step 4.1.11.4

Add $1$ and $1$ .

$\frac{7 \sqrt{3} - 21 i + 7 i + \sqrt{7} (- i^{2} \sqrt{21})}{4}$

Step 4.1.11.5

Combine using the product rule for radicals.

$\frac{7 \sqrt{3} - 21 i + 7 i - i^{2} \sqrt{7 \cdot 21}}{4}$

Step 4.1.11.6

Multiply $7$ by $21$ .

$\frac{7 \sqrt{3} - 21 i + 7 i - i^{2} \sqrt{147}}{4}$

Step 4.1.12

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

$\frac{7 \sqrt{3} - 21 i + 7 i - - 1 \sqrt{147}}{4}$

Step 4.1.13

Multiply $- 1$ by $- 1$ .

$\frac{7 \sqrt{3} - 21 i + 7 i + 1 \sqrt{147}}{4}$

Step 4.1.14

Multiply $\sqrt{147}$ by $1$ .

$\frac{7 \sqrt{3} - 21 i + 7 i + \sqrt{147}}{4}$

Step 4.1.15

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 4.1.15.1

Factor $49$ out of $147$ .

$\frac{7 \sqrt{3} - 21 i + 7 i + \sqrt{49 (3)}}{4}$

Step 4.1.15.2

Rewrite $49$ as $7^{2}$ .

$\frac{7 \sqrt{3} - 21 i + 7 i + \sqrt{7^{2} \cdot 3}}{4}$

Step 4.1.16

Pull terms out from under the radical.

$\frac{7 \sqrt{3} - 21 i + 7 i + 7 \sqrt{3}}{4}$

Step 4.2

Add $7 \sqrt{3}$ and $7 \sqrt{3}$ .

$\frac{- 21 i + 7 i + 14 \sqrt{3}}{4}$

Step 4.3

Add $- 21 i$ and $7 i$ .

$\frac{- 14 i + 14 \sqrt{3}}{4}$

Step 5

Reorder $- 14 i$ and $14 \sqrt{3}$ .

$\frac{14 \sqrt{3} - 14 i}{4}$

Step 6

Cancel the common factor of $14 \sqrt{3} - 14 i$ and $4$ .

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Step 6.1

Factor $2$ out of $14 \sqrt{3}$ .

$\frac{2 (7 \sqrt{3}) - 14 i}{4}$

Step 6.2

Factor $2$ out of $- 14 i$ .

$\frac{2 (7 \sqrt{3}) + 2 (- 7 i)}{4}$

Step 6.3

Factor $2$ out of $2 (7 \sqrt{3}) + 2 (- 7 i)$ .

$\frac{2 (7 \sqrt{3} - 7 i)}{4}$

Step 6.4

Cancel the common factors.

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Step 6.4.1

Factor $2$ out of $4$ .

$\frac{2 (7 \sqrt{3} - 7 i)}{2 (2)}$

Step 6.4.2

Cancel the common factor.

$\frac{2 (7 \sqrt{3} - 7 i)}{2 \cdot 2}$

Step 6.4.3

Rewrite the expression.

$\frac{7 \sqrt{3} - 7 i}{2}$