Simplify (10+ square root of -200)(4+ square root of -200)

$(10 + \sqrt{- 200}) (4 + \sqrt{- 200})$

Rewrite $- 200$ as $- 1 (200)$ .

$(10 + \sqrt{- 1 (200)}) (4 + \sqrt{- 200})$

Rewrite $\sqrt{- 1 (200)}$ as $\sqrt{- 1} \cdot \sqrt{200}$ .

$(10 + \sqrt{- 1} \cdot \sqrt{200}) (4 + \sqrt{- 200})$

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

$(10 + i \cdot \sqrt{200}) (4 + \sqrt{- 200})$

Rewrite $- 200$ as $- 1 (200)$ .

$(10 + i \cdot \sqrt{200}) (4 + \sqrt{- 1 (200)})$

Rewrite $\sqrt{- 1 (200)}$ as $\sqrt{- 1} \cdot \sqrt{200}$ .

$(10 + i \cdot \sqrt{200}) (4 + \sqrt{- 1} \cdot \sqrt{200})$

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

$(10 + i \cdot \sqrt{200}) (4 + i \cdot \sqrt{200})$

Simplify terms.

Simplify each term.

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$(10 + i \cdot \sqrt{100 (2)}) (4 + i \cdot \sqrt{200})$

Rewrite $100$ as $10^{2}$ .

$(10 + i \cdot \sqrt{10^{2} \cdot 2}) (4 + i \cdot \sqrt{200})$

Pull terms out from under the radical.

$(10 + i \cdot (10 \sqrt{2})) (4 + i \cdot \sqrt{200})$

Move $10$ to the left of $i$ .

$(10 + 10 i \sqrt{2}) (4 + i \cdot \sqrt{200})$

Simplify each term.

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$(10 + 10 i \sqrt{2}) (4 + i \cdot \sqrt{100 (2)})$

Rewrite $100$ as $10^{2}$ .

$(10 + 10 i \sqrt{2}) (4 + i \cdot \sqrt{10^{2} \cdot 2})$

Pull terms out from under the radical.

$(10 + 10 i \sqrt{2}) (4 + i \cdot (10 \sqrt{2}))$

Move $10$ to the left of $i$ .

$(10 + 10 i \sqrt{2}) (4 + 10 i \sqrt{2})$

Expand $(10 + 10 i \sqrt{2}) (4 + 10 i \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$10 (4 + 10 i \sqrt{2}) + 10 i \sqrt{2} (4 + 10 i \sqrt{2})$

Apply the distributive property.

$10 \cdot 4 + 10 (10 i \sqrt{2}) + 10 i \sqrt{2} (4 + 10 i \sqrt{2})$

Apply the distributive property.

$10 \cdot 4 + 10 (10 i \sqrt{2}) + 10 i \sqrt{2} \cdot 4 + 10 i \sqrt{2} (10 i \sqrt{2})$

Simplify and combine like terms.

Simplify each term.

Multiply $10$ by $4$ .

$40 + 10 (10 i \sqrt{2}) + 10 i \sqrt{2} \cdot 4 + 10 i \sqrt{2} (10 i \sqrt{2})$

Multiply $10$ by $10$ .

$40 + 100 (i \sqrt{2}) + 10 i \sqrt{2} \cdot 4 + 10 i \sqrt{2} (10 i \sqrt{2})$

Multiply $4$ by $10$ .

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 10 i \sqrt{2} (10 i \sqrt{2})$

Multiply $10 i \sqrt{2} (10 i \sqrt{2})$ .

Multiply $10$ by $10$ .

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 i \sqrt{2} (i \sqrt{2})$

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

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 (i^{1} i) \sqrt{2} \sqrt{2}$

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

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 (i^{1} i^{1}) \sqrt{2} \sqrt{2}$

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

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 i^{1 + 1} \sqrt{2} \sqrt{2}$

Add $1$ and $1$ .

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 i^{2} \sqrt{2} \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 i^{2} ({\sqrt{2}}^{1} \sqrt{2})$

Raise $\sqrt{2}$ to the power of $1$ .

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

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

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 i^{2} {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 i^{2} {\sqrt{2}}^{2}$

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

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} + 100 \cdot - 1 {\sqrt{2}}^{2}$

Multiply $100$ by $- 1$ .

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} - 100 {\sqrt{2}}^{2}$

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

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

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} - 100 {(2^{\frac{1}{2}})}^{2}$

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

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} - 100 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} - 100 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} - 100 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} - 100 \cdot 2^{1}$

Evaluate the exponent.

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} - 100 \cdot 2$

Multiply $- 100$ by $2$ .

$40 + 100 i \sqrt{2} + 40 i \sqrt{2} - 200$

Subtract $200$ from $40$ .

$100 i \sqrt{2} + 40 i \sqrt{2} - 160$

Add $100 i \sqrt{2}$ and $40 i \sqrt{2}$ .

$140 i \sqrt{2} - 160$

Reorder $140 i \sqrt{2}$ and $- 160$ .

$- 160 + 140 i \sqrt{2}$

Simplify (10+ square root of -200)(4+ square root of -200)

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