Algebra Examples

$(6 + \sqrt{- 3}) (6 + \sqrt{- 7})$

Step 1

Rewrite $- 3$ as $- 1 (3)$ .

$(6 + \sqrt{- 1 (3)}) (6 + \sqrt{- 7})$

Step 2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$(6 + \sqrt{- 1} \cdot \sqrt{3}) (6 + \sqrt{- 7})$

Step 3

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

$(6 + i \cdot \sqrt{3}) (6 + \sqrt{- 7})$

Step 4

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

$(6 + i \cdot \sqrt{3}) (6 + \sqrt{- 1 (7)})$

Step 5

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

$(6 + i \cdot \sqrt{3}) (6 + \sqrt{- 1} \cdot \sqrt{7})$

Step 6

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

$(6 + i \cdot \sqrt{3}) (6 + i \cdot \sqrt{7})$

Step 7

Expand $(6 + i \sqrt{3}) (6 + i \sqrt{7})$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$6 (6 + i \sqrt{7}) + i \sqrt{3} (6 + i \sqrt{7})$

Step 7.2

Apply the distributive property.

$6 \cdot 6 + 6 (i \sqrt{7}) + i \sqrt{3} (6 + i \sqrt{7})$

Step 7.3

Apply the distributive property.

$6 \cdot 6 + 6 (i \sqrt{7}) + i \sqrt{3} \cdot 6 + i \sqrt{3} (i \sqrt{7})$

Step 8

Simplify terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $6$ by $6$ .

$36 + 6 (i \sqrt{7}) + i \sqrt{3} \cdot 6 + i \sqrt{3} (i \sqrt{7})$

Step 8.1.2

Move $6$ to the left of $i \sqrt{3}$ .

$36 + 6 i \sqrt{7} + 6 \cdot (i \sqrt{3}) + i \sqrt{3} (i \sqrt{7})$

Step 8.1.3

Multiply $i \sqrt{3} (i \sqrt{7})$ .

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Step 8.1.3.1

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

$36 + 6 i \sqrt{7} + 6 i \sqrt{3} + i^{1} i \sqrt{3} \sqrt{7}$

Step 8.1.3.2

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

$36 + 6 i \sqrt{7} + 6 i \sqrt{3} + i^{1} i^{1} \sqrt{3} \sqrt{7}$

Step 8.1.3.3

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

$36 + 6 i \sqrt{7} + 6 i \sqrt{3} + i^{1 + 1} \sqrt{3} \sqrt{7}$

Step 8.1.3.4

Add $1$ and $1$ .

$36 + 6 i \sqrt{7} + 6 i \sqrt{3} + i^{2} \sqrt{3} \sqrt{7}$

Step 8.1.3.5

Combine using the product rule for radicals.

$36 + 6 i \sqrt{7} + 6 i \sqrt{3} + i^{2} \sqrt{7 \cdot 3}$

Step 8.1.3.6

Multiply $7$ by $3$ .

$36 + 6 i \sqrt{7} + 6 i \sqrt{3} + i^{2} \sqrt{21}$

Step 8.1.4

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

$36 + 6 i \sqrt{7} + 6 i \sqrt{3} - 1 \sqrt{21}$

Step 8.1.5

Rewrite $- 1 \sqrt{21}$ as $- \sqrt{21}$ .

$36 + 6 i \sqrt{7} + 6 i \sqrt{3} - \sqrt{21}$

Step 8.2

Reorder $36 + 6 i \sqrt{7} + 6 i \sqrt{3}$ and $- \sqrt{21}$ .

$- \sqrt{21} + 36 + 6 i \sqrt{7} + 6 i \sqrt{3}$