Algebra Examples

$(- 3 + 2 i \sqrt{48}) (- 5 - 4 i \sqrt{12})$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 1.1.1.1

Factor $16$ out of $48$ .

$(- 3 + 2 i \sqrt{16 (3)}) (- 5 - 4 i \sqrt{12})$

Step 1.1.1.2

Rewrite $16$ as $4^{2}$ .

$(- 3 + 2 i \sqrt{4^{2} \cdot 3}) (- 5 - 4 i \sqrt{12})$

Step 1.1.2

Pull terms out from under the radical.

$(- 3 + 2 i (4 \sqrt{3})) (- 5 - 4 i \sqrt{12})$

Step 1.1.3

Multiply $4$ by $2$ .

$(- 3 + 8 i \sqrt{3}) (- 5 - 4 i \sqrt{12})$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.1.1

Factor $4$ out of $12$ .

$(- 3 + 8 i \sqrt{3}) (- 5 - 4 i \sqrt{4 (3)})$

Step 1.2.1.2

Rewrite $4$ as $2^{2}$ .

$(- 3 + 8 i \sqrt{3}) (- 5 - 4 i \sqrt{2^{2} \cdot 3})$

Step 1.2.2

Pull terms out from under the radical.

$(- 3 + 8 i \sqrt{3}) (- 5 - 4 i (2 \sqrt{3}))$

Step 1.2.3

Multiply $2$ by $- 4$ .

$(- 3 + 8 i \sqrt{3}) (- 5 - 8 i \sqrt{3})$

Step 2

Expand $(- 3 + 8 i \sqrt{3}) (- 5 - 8 i \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$- 3 (- 5 - 8 i \sqrt{3}) + 8 i \sqrt{3} (- 5 - 8 i \sqrt{3})$

Step 2.2

Apply the distributive property.

$- 3 \cdot - 5 - 3 (- 8 i \sqrt{3}) + 8 i \sqrt{3} (- 5 - 8 i \sqrt{3})$

Step 2.3

Apply the distributive property.

$- 3 \cdot - 5 - 3 (- 8 i \sqrt{3}) + 8 i \sqrt{3} \cdot - 5 + 8 i \sqrt{3} (- 8 i \sqrt{3})$

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 $- 3$ by $- 5$ .

$15 - 3 (- 8 i \sqrt{3}) + 8 i \sqrt{3} \cdot - 5 + 8 i \sqrt{3} (- 8 i \sqrt{3})$

Step 3.1.2

Multiply $- 8$ by $- 3$ .

$15 + 24 (i \sqrt{3}) + 8 i \sqrt{3} \cdot - 5 + 8 i \sqrt{3} (- 8 i \sqrt{3})$

Step 3.1.3

Multiply $- 5$ by $8$ .

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 8 i \sqrt{3} (- 8 i \sqrt{3})$

Step 3.1.4

Multiply $8 i \sqrt{3} (- 8 i \sqrt{3})$ .

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Step 3.1.4.1

Multiply $- 8$ by $8$ .

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 i \sqrt{3} (i \sqrt{3})$

Step 3.1.4.2

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 (i^{1} i) \sqrt{3} \sqrt{3}$

Step 3.1.4.3

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 (i^{1} i^{1}) \sqrt{3} \sqrt{3}$

Step 3.1.4.4

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 i^{1 + 1} \sqrt{3} \sqrt{3}$

Step 3.1.4.5

Add $1$ and $1$ .

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 i^{2} \sqrt{3} \sqrt{3}$

Step 3.1.4.6

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Step 3.1.4.7

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 3.1.4.8

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 i^{2} {\sqrt{3}}^{1 + 1}$

Step 3.1.4.9

Add $1$ and $1$ .

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 i^{2} {\sqrt{3}}^{2}$

Step 3.1.5

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} - 64 \cdot - 1 {\sqrt{3}}^{2}$

Step 3.1.6

Multiply $- 64$ by $- 1$ .

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 64 {\sqrt{3}}^{2}$

Step 3.1.7

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

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Step 3.1.7.1

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 64 {(3^{\frac{1}{2}})}^{2}$

Step 3.1.7.2

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 64 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 3.1.7.3

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

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 64 \cdot 3^{\frac{2}{2}}$

Step 3.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.7.4.1

Cancel the common factor.

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 64 \cdot 3^{\frac{2}{2}}$

Step 3.1.7.4.2

Rewrite the expression.

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 64 \cdot 3^{1}$

Step 3.1.7.5

Evaluate the exponent.

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 64 \cdot 3$

Step 3.1.8

Multiply $64$ by $3$ .

$15 + 24 i \sqrt{3} - 40 i \sqrt{3} + 192$

Step 3.2

Add $15$ and $192$ .

$24 i \sqrt{3} - 40 i \sqrt{3} + 207$

Step 3.3

Subtract $40 i \sqrt{3}$ from $24 i \sqrt{3}$ .

$- 16 i \sqrt{3} + 207$

Step 4

Reorder $- 16 i \sqrt{3}$ and $207$ .

$207 - 16 i \sqrt{3}$