Algebra Examples

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

Step 1

Rewrite ${(i - \sqrt{3})}^{2}$ as $(i - \sqrt{3}) (i - \sqrt{3})$ .

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

Step 2

Expand $(i - \sqrt{3}) (i - \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $i i$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

Add $1$ and $1$ .

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.4.4.1

Cancel the common factor.

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

Step 3.1.4.4.2

Rewrite the expression.

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

Step 3.1.4.5

Evaluate the exponent.

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

Step 3.2

Add $- 1$ and $3$ .

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

Step 3.3

Reorder the factors of $- \sqrt{3} i$ .

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

Step 3.4

Subtract $i \sqrt{3}$ from $i (- \sqrt{3})$ .

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Step 3.4.1

Reorder $i$ and $- 1$ .

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

Step 3.4.2

Subtract $i \sqrt{3}$ from $- 1 \cdot i \sqrt{3}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify terms.

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Step 6.1

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

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

Step 6.2

Expand the denominator using the FOIL method.

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

Step 6.3

Simplify.

$\frac{(3 + i \sqrt{3}) (2 + 2 i \sqrt{3})}{16}$

Step 6.4

Cancel the common factor of $2 + 2 i \sqrt{3}$ and $16$ .

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Step 6.4.1

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

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

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $2$ out of $16$ .

$\frac{2 ((3 + i \sqrt{3}) (1 + i \sqrt{3}))}{2 (8)}$

Step 6.4.2.2

Cancel the common factor.

$\frac{2 ((3 + i \sqrt{3}) (1 + i \sqrt{3}))}{2 \cdot 8}$

Step 6.4.2.3

Rewrite the expression.

$\frac{(3 + i \sqrt{3}) (1 + i \sqrt{3})}{8}$

Step 7

Expand $(3 + i \sqrt{3}) (1 + i \sqrt{3})$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$\frac{3 (1 + i \sqrt{3}) + i \sqrt{3} (1 + i \sqrt{3})}{8}$

Step 7.2

Apply the distributive property.

$\frac{3 \cdot 1 + 3 (i \sqrt{3}) + i \sqrt{3} (1 + i \sqrt{3})}{8}$

Step 7.3

Apply the distributive property.

$\frac{3 \cdot 1 + 3 (i \sqrt{3}) + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{8}$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $3$ by $1$ .

$\frac{3 + 3 (i \sqrt{3}) + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{8}$

Step 8.1.2

Multiply $i$ by $1$ .

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i \sqrt{3} (i \sqrt{3})}{8}$

Step 8.1.3

Multiply $i \sqrt{3} (i \sqrt{3})$ .

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Step 8.1.3.1

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i^{1} i \sqrt{3} \sqrt{3}}{8}$

Step 8.1.3.2

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i^{1} i^{1} \sqrt{3} \sqrt{3}}{8}$

Step 8.1.3.3

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i^{1 + 1} \sqrt{3} \sqrt{3}}{8}$

Step 8.1.3.4

Add $1$ and $1$ .

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i^{2} \sqrt{3} \sqrt{3}}{8}$

Step 8.1.3.5

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{8}$

Step 8.1.3.6

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{8}$

Step 8.1.3.7

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i^{2} {\sqrt{3}}^{1 + 1}}{8}$

Step 8.1.3.8

Add $1$ and $1$ .

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} + i^{2} {\sqrt{3}}^{2}}{8}$

Step 8.1.4

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} - 1 {\sqrt{3}}^{2}}{8}$

Step 8.1.5

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

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Step 8.1.5.1

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

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

Step 8.1.5.2

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{1}{2} \cdot 2}}{8}$

Step 8.1.5.3

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

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}}{8}$

Step 8.1.5.4

Cancel the common factor of $2$ .

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Step 8.1.5.4.1

Cancel the common factor.

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}}{8}$

Step 8.1.5.4.2

Rewrite the expression.

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{1}}{8}$

Step 8.1.5.5

Evaluate the exponent.

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} - 1 \cdot 3}{8}$

Step 8.1.6

Multiply $- 1$ by $3$ .

$\frac{3 + 3 i \sqrt{3} + i \sqrt{3} - 3}{8}$

Step 8.2

Subtract $3$ from $3$ .

$\frac{3 i \sqrt{3} + i \sqrt{3} + 0}{8}$

Step 8.3

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

$\frac{4 i \sqrt{3} + 0}{8}$

Step 8.4

Add $4 i \sqrt{3}$ and $0$ .

$\frac{4 i \sqrt{3}}{8}$

Step 9

Cancel the common factor of $4$ and $8$ .

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Step 9.1

Factor $4$ out of $4 i \sqrt{3}$ .

$\frac{4 (i \sqrt{3})}{8}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $4$ out of $8$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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