Finite Math Examples

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

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 7$ as $- 1 (7)$ .

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

Step 1.2

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

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

Step 1.3

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 2

Multiply $\frac{2 i + 3 \sqrt{3}}{3 i \cdot \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{2 i + 3 \sqrt{3}}{3 i \cdot \sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{2 i + 3 \sqrt{3}}{3 i \cdot \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

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

Step 3.2

Move $\sqrt{7}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Evaluate the exponent.

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

Step 4

Multiply $7$ by $3$ .

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

Step 5

Apply the distributive property.

$\frac{2 i \sqrt{7} + 3 \sqrt{3} \sqrt{7}}{21 i}$

Step 6

Multiply $3 \sqrt{3} \sqrt{7}$ .

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $7$ by $3$ .

$\frac{2 i \sqrt{7} + 3 \sqrt{21}}{21 i}$

Step 7

Reorder $2 i \sqrt{7}$ and $3 \sqrt{21}$ .

$\frac{3 \sqrt{21} + 2 i \sqrt{7}}{21 i}$