Simplify (-2+3i square root of 2)(5-4i square root of 2)

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

Step 1

Expand $(- 2 + 3 i \sqrt{2}) (5 - 4 i \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $5$ .

$- 10 - 2 (- 4 i \sqrt{2}) + 3 i \sqrt{2} \cdot 5 + 3 i \sqrt{2} (- 4 i \sqrt{2})$

Multiply $- 4$ by $- 2$ .

$- 10 + 8 (i \sqrt{2}) + 3 i \sqrt{2} \cdot 5 + 3 i \sqrt{2} (- 4 i \sqrt{2})$

Multiply $5$ by $3$ .

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 3 i \sqrt{2} (- 4 i \sqrt{2})$

Multiply $3 i \sqrt{2} (- 4 i \sqrt{2})$ .

Multiply $- 4$ by $3$ .

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 i \sqrt{2} (i \sqrt{2})$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 (i^{1} i) \sqrt{2} \sqrt{2}$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 (i^{1} i^{1}) \sqrt{2} \sqrt{2}$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 i^{1 + 1} \sqrt{2} \sqrt{2}$

Add $1$ and $1$ .

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 i^{2} \sqrt{2} \sqrt{2}$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 i^{2} ({\sqrt{2}}^{1} \sqrt{2})$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 i^{2} {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 i^{2} {\sqrt{2}}^{2}$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} - 12 \cdot - 1 {\sqrt{2}}^{2}$

Multiply $- 12$ by $- 1$ .

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 12 {\sqrt{2}}^{2}$

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

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 12 {(2^{\frac{1}{2}})}^{2}$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 12 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 12 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 12 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 12 \cdot 2^{1}$

Evaluate the exponent.

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 12 \cdot 2$

Multiply $12$ by $2$ .

$- 10 + 8 i \sqrt{2} + 15 i \sqrt{2} + 24$

Add $- 10$ and $24$ .

$8 i \sqrt{2} + 15 i \sqrt{2} + 14$

Add $8 i \sqrt{2}$ and $15 i \sqrt{2}$ .

$23 i \sqrt{2} + 14$

Step 3

Reorder $23 i \sqrt{2}$ and $14$ .

$14 + 23 i \sqrt{2}$

Simplify (-2+3i square root of 2)(5-4i square root of 2)

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