Simplify (i square root of 2-1)(2i square root of 2+3)

$(i \sqrt{2} - 1) (2 i \sqrt{2} + 3)$

Step 1

Expand $(i \sqrt{2} - 1) (2 i \sqrt{2} + 3)$ using the FOIL Method.

Apply the distributive property.

$i \sqrt{2} (2 i \sqrt{2} + 3) - 1 (2 i \sqrt{2} + 3)$

Apply the distributive property.

$i \sqrt{2} (2 i \sqrt{2}) + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2} + 3)$

Apply the distributive property.

$i \sqrt{2} (2 i \sqrt{2}) + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Step 2

Simplify and combine like terms.

Simplify each term.

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

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

$\sqrt{2} (2 (i^{1} i) \sqrt{2}) + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

$\sqrt{2} (2 (i^{1} i^{1}) \sqrt{2}) + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

$\sqrt{2} (2 i^{1 + 1} \sqrt{2}) + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Add $1$ and $1$ .

$\sqrt{2} (2 i^{2} \sqrt{2}) + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

$2 i^{2} ({\sqrt{2}}^{1} \sqrt{2}) + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

$2 i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

$2 i^{2} {\sqrt{2}}^{1 + 1} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Add $1$ and $1$ .

$2 i^{2} {\sqrt{2}}^{2} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

$2 \cdot - 1 {\sqrt{2}}^{2} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Multiply $2$ by $- 1$ .

$- 2 {\sqrt{2}}^{2} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

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

$- 2 {(2^{\frac{1}{2}})}^{2} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

$- 2 \cdot 2^{\frac{1}{2} \cdot 2} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

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

$- 2 \cdot 2^{\frac{2}{2}} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 2 \cdot 2^{\frac{2}{2}} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Rewrite the expression.

$- 2 \cdot 2^{1} + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Evaluate the exponent.

$- 2 \cdot 2 + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Multiply $- 2$ by $2$ .

$- 4 + i \sqrt{2} \cdot 3 - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Move $3$ to the left of $i \sqrt{2}$ .

$- 4 + 3 \cdot (i \sqrt{2}) - 1 (2 i \sqrt{2}) - 1 \cdot 3$

Multiply $2$ by $- 1$ .

$- 4 + 3 i \sqrt{2} - 2 (i \sqrt{2}) - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$- 4 + 3 i \sqrt{2} - 2 i \sqrt{2} - 3$

Subtract $3$ from $- 4$ .

$3 i \sqrt{2} - 2 i \sqrt{2} - 7$

Subtract $2 i \sqrt{2}$ from $3 i \sqrt{2}$ .

$1 i \sqrt{2} - 7$

Step 3

Multiply $i$ by $1$ .

$i \sqrt{2} - 7$

Step 4

Reorder $i \sqrt{2}$ and $- 7$ .

$- 7 + i \sqrt{2}$

Simplify (i square root of 2-1)(2i square root of 2+3)

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