Algebra Examples

${(1 + \sqrt{3} i)}^{2} = a + i b$

Step 1

Rewrite the equation as $a + i b = {(1 + \sqrt{3} i)}^{2}$ .

$a + i b = {(1 + \sqrt{3} i)}^{2}$

Step 2

Simplify ${(1 + \sqrt{3} i)}^{2}$ .

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Step 2.1

Rewrite.

$a + i b = 0 + 0 + {(1 + \sqrt{3} i)}^{2}$

Step 2.2

Rewrite ${(1 + \sqrt{3} i)}^{2}$ as $(1 + \sqrt{3} i) (1 + \sqrt{3} i)$ .

$a + i b = (1 + \sqrt{3} i) (1 + \sqrt{3} i)$

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

$a + i b = 1 (1 + \sqrt{3} i) + \sqrt{3} i (1 + \sqrt{3} i)$

Step 2.3.2

Apply the distributive property.

$a + i b = 1 \cdot 1 + 1 (\sqrt{3} i) + \sqrt{3} i (1 + \sqrt{3} i)$

Step 2.3.3

Apply the distributive property.

$a + i b = 1 \cdot 1 + 1 (\sqrt{3} i) + \sqrt{3} i \cdot 1 + \sqrt{3} i (\sqrt{3} i)$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $1$ by $1$ .

$a + i b = 1 + 1 (\sqrt{3} i) + \sqrt{3} i \cdot 1 + \sqrt{3} i (\sqrt{3} i)$

Step 2.4.1.2

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i \cdot 1 + \sqrt{3} i (\sqrt{3} i)$

Step 2.4.1.3

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + \sqrt{3} i (\sqrt{3} i)$

Step 2.4.1.4

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

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Step 2.4.1.4.1

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1} \sqrt{3} i i$

Step 2.4.1.4.2

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1} {\sqrt{3}}^{1} i i$

Step 2.4.1.4.3

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1 + 1} i i$

Step 2.4.1.4.4

Add $1$ and $1$ .

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i i$

Step 2.4.1.4.5

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} (i^{1} i)$

Step 2.4.1.4.6

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} (i^{1} i^{1})$

Step 2.4.1.4.7

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i^{1 + 1}$

Step 2.4.1.4.8

Add $1$ and $1$ .

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i^{2}$

Step 2.4.1.5

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

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Step 2.4.1.5.1

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + {(3^{\frac{1}{2}})}^{2} i^{2}$

Step 2.4.1.5.2

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{1}{2} \cdot 2} i^{2}$

Step 2.4.1.5.3

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{2}{2}} i^{2}$

Step 2.4.1.5.4

Cancel the common factor of $2$ .

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Step 2.4.1.5.4.1

Cancel the common factor.

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{2}{2}} i^{2}$

Step 2.4.1.5.4.2

Rewrite the expression.

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + 3^{1} i^{2}$

Step 2.4.1.5.5

Evaluate the exponent.

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + 3 i^{2}$

Step 2.4.1.6

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

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i + 3 \cdot - 1$

Step 2.4.1.7

Multiply $3$ by $- 1$ .

$a + i b = 1 + \sqrt{3} i + \sqrt{3} i - 3$

Step 2.4.2

Subtract $3$ from $1$ .

$a + i b = - 2 + \sqrt{3} i + \sqrt{3} i$

Step 2.4.3

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

$a + i b = - 2 + 2 \sqrt{3} i$

Step 3

Subtract $a$ from both sides of the equation.

$i b = - 2 + 2 \sqrt{3} i - a$

Step 4

Divide each term in $i b = - 2 + 2 \sqrt{3} i - a$ by $i$ and simplify.

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Step 4.1

Divide each term in $i b = - 2 + 2 \sqrt{3} i - a$ by $i$ .

$\frac{i b}{i} = \frac{- 2}{i} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $i$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{i b}{i} = \frac{- 2}{i} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 2}{i} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply the numerator and denominator of $\frac{- 2}{i}$ by the conjugate of $i$ to make the denominator real.

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

Step 4.3.1.2

Multiply.

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Step 4.3.1.2.1

Combine.

$b = \frac{- 2 i}{i i} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2

Simplify the denominator.

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Step 4.3.1.2.2.1

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

$b = \frac{- 2 i}{i^{1} i} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2.2

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

$b = \frac{- 2 i}{i^{1} i^{1}} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2.3

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

$b = \frac{- 2 i}{i^{1 + 1}} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2.4

Add $1$ and $1$ .

$b = \frac{- 2 i}{i^{2}} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2.5

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

$b = \frac{- 2 i}{- 1} + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.3

Move the negative one from the denominator of $\frac{- 2 i}{- 1}$ .

$b = - 1 \cdot (- 2 i) + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.4

Rewrite $- 1 \cdot (- 2 i)$ as $- (- 2 i)$ .

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

Step 4.3.1.5

Multiply $- 2$ by $- 1$ .

$b = 2 i + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.6

Cancel the common factor of $i$ .

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Step 4.3.1.6.1

Cancel the common factor.

$b = 2 i + \frac{2 \sqrt{3} i}{i} + \frac{- a}{i}$

Step 4.3.1.6.2

Divide $2 \sqrt{3}$ by $1$ .

$b = 2 i + 2 \sqrt{3} + \frac{- a}{i}$

Step 4.3.1.7

Multiply the numerator and denominator of $\frac{- a}{i}$ by the conjugate of $i$ to make the denominator real.

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

Step 4.3.1.8

Multiply.

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Step 4.3.1.8.1

Combine.

$b = 2 i + 2 \sqrt{3} + \frac{- a i}{i i}$

Step 4.3.1.8.2

Simplify the denominator.

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Step 4.3.1.8.2.1

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

$b = 2 i + 2 \sqrt{3} + \frac{- a i}{i^{1} i}$

Step 4.3.1.8.2.2

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

$b = 2 i + 2 \sqrt{3} + \frac{- a i}{i^{1} i^{1}}$

Step 4.3.1.8.2.3

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

$b = 2 i + 2 \sqrt{3} + \frac{- a i}{i^{1 + 1}}$

Step 4.3.1.8.2.4

Add $1$ and $1$ .

$b = 2 i + 2 \sqrt{3} + \frac{- a i}{i^{2}}$

Step 4.3.1.8.2.5

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

$b = 2 i + 2 \sqrt{3} + \frac{- a i}{- 1}$

Step 4.3.1.9

Dividing two negative values results in a positive value.

$b = 2 i + 2 \sqrt{3} + \frac{a i}{1}$

Step 4.3.1.10

Divide $a i$ by $1$ .

$b = 2 i + 2 \sqrt{3} + a i$

Step 4.3.2

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

$b = 2 \sqrt{3} + 2 i + a i$