Simplify (13+ square root of -180)(6+ square root of -180)

$(13 + \sqrt{- 180}) (6 + \sqrt{- 180})$

Step 1

Rewrite $- 180$ as $- 1 (180)$ .

$(13 + \sqrt{- 1 (180)}) (6 + \sqrt{- 180})$

Step 2

Rewrite $\sqrt{- 1 (180)}$ as $\sqrt{- 1} \cdot \sqrt{180}$ .

$(13 + \sqrt{- 1} \cdot \sqrt{180}) (6 + \sqrt{- 180})$

Step 3

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

$(13 + i \cdot \sqrt{180}) (6 + \sqrt{- 180})$

Step 4

Rewrite $- 180$ as $- 1 (180)$ .

$(13 + i \cdot \sqrt{180}) (6 + \sqrt{- 1 (180)})$

Step 5

Rewrite $\sqrt{- 1 (180)}$ as $\sqrt{- 1} \cdot \sqrt{180}$ .

$(13 + i \cdot \sqrt{180}) (6 + \sqrt{- 1} \cdot \sqrt{180})$

Step 6

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

$(13 + i \cdot \sqrt{180}) (6 + i \cdot \sqrt{180})$

Step 7

Simplify terms.

Simplify each term.

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$(13 + i \cdot \sqrt{36 (5)}) (6 + i \cdot \sqrt{180})$

Rewrite $36$ as $6^{2}$ .

$(13 + i \cdot \sqrt{6^{2} \cdot 5}) (6 + i \cdot \sqrt{180})$

Pull terms out from under the radical.

$(13 + i \cdot (6 \sqrt{5})) (6 + i \cdot \sqrt{180})$

Move $6$ to the left of $i$ .

$(13 + 6 i \sqrt{5}) (6 + i \cdot \sqrt{180})$

Simplify each term.

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$(13 + 6 i \sqrt{5}) (6 + i \cdot \sqrt{36 (5)})$

Rewrite $36$ as $6^{2}$ .

$(13 + 6 i \sqrt{5}) (6 + i \cdot \sqrt{6^{2} \cdot 5})$

Pull terms out from under the radical.

$(13 + 6 i \sqrt{5}) (6 + i \cdot (6 \sqrt{5}))$

Move $6$ to the left of $i$ .

$(13 + 6 i \sqrt{5}) (6 + 6 i \sqrt{5})$

Step 8

Expand $(13 + 6 i \sqrt{5}) (6 + 6 i \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$13 (6 + 6 i \sqrt{5}) + 6 i \sqrt{5} (6 + 6 i \sqrt{5})$

Apply the distributive property.

$13 \cdot 6 + 13 (6 i \sqrt{5}) + 6 i \sqrt{5} (6 + 6 i \sqrt{5})$

Apply the distributive property.

$13 \cdot 6 + 13 (6 i \sqrt{5}) + 6 i \sqrt{5} \cdot 6 + 6 i \sqrt{5} (6 i \sqrt{5})$

Step 9

Simplify and combine like terms.

Simplify each term.

Multiply $13$ by $6$ .

$78 + 13 (6 i \sqrt{5}) + 6 i \sqrt{5} \cdot 6 + 6 i \sqrt{5} (6 i \sqrt{5})$

Multiply $6$ by $13$ .

$78 + 78 (i \sqrt{5}) + 6 i \sqrt{5} \cdot 6 + 6 i \sqrt{5} (6 i \sqrt{5})$

Multiply $6$ by $6$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 6 i \sqrt{5} (6 i \sqrt{5})$

Multiply $6 i \sqrt{5} (6 i \sqrt{5})$ .

Multiply $6$ by $6$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 i \sqrt{5} (i \sqrt{5})$

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

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 (i^{1} i) \sqrt{5} \sqrt{5}$

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

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 (i^{1} i^{1}) \sqrt{5} \sqrt{5}$

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

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 i^{1 + 1} \sqrt{5} \sqrt{5}$

Add $1$ and $1$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 i^{2} \sqrt{5} \sqrt{5}$

Raise $\sqrt{5}$ to the power of $1$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 i^{2} ({\sqrt{5}}^{1} \sqrt{5})$

Raise $\sqrt{5}$ to the power of $1$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 i^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

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

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 i^{2} {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 i^{2} {\sqrt{5}}^{2}$

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

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} + 36 \cdot - 1 {\sqrt{5}}^{2}$

Multiply $36$ by $- 1$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} - 36 {\sqrt{5}}^{2}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} - 36 {(5^{\frac{1}{2}})}^{2}$

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

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} - 36 \cdot 5^{\frac{1}{2} \cdot 2}$

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

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} - 36 \cdot 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} - 36 \cdot 5^{\frac{2}{2}}$

Rewrite the expression.

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} - 36 \cdot 5^{1}$

Evaluate the exponent.

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} - 36 \cdot 5$

Multiply $- 36$ by $5$ .

$78 + 78 i \sqrt{5} + 36 i \sqrt{5} - 180$

Subtract $180$ from $78$ .

$78 i \sqrt{5} + 36 i \sqrt{5} - 102$

Add $78 i \sqrt{5}$ and $36 i \sqrt{5}$ .

$114 i \sqrt{5} - 102$

Step 10

Reorder $114 i \sqrt{5}$ and $- 102$ .

$- 102 + 114 i \sqrt{5}$

Simplify (13+ square root of -180)(6+ square root of -180)

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