Finite Math Examples

$(\sqrt{5} - 4 i) (\sqrt{5} + 7 i)$

Step 1

Expand $(\sqrt{5} - 4 i) (\sqrt{5} + 7 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{5} (\sqrt{5} + 7 i) - 4 i (\sqrt{5} + 7 i)$

Step 1.2

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} (7 i) - 4 i (\sqrt{5} + 7 i)$

Step 1.3

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} (7 i) - 4 i \sqrt{5} - 4 i (7 i)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

$\sqrt{5 \cdot 5} + \sqrt{5} (7 i) - 4 i \sqrt{5} - 4 i (7 i)$

Step 2.1.2

Multiply $5$ by $5$ .

$\sqrt{25} + \sqrt{5} (7 i) - 4 i \sqrt{5} - 4 i (7 i)$

Step 2.1.3

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}} + \sqrt{5} (7 i) - 4 i \sqrt{5} - 4 i (7 i)$

Step 2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$5 + \sqrt{5} (7 i) - 4 i \sqrt{5} - 4 i (7 i)$

Step 2.1.5

Move $7$ to the left of $\sqrt{5}$ .

$5 + 7 \cdot \sqrt{5} i - 4 i \sqrt{5} - 4 i (7 i)$

Step 2.1.6

Multiply $- 4 i (7 i)$ .

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Step 2.1.6.1

Multiply $7$ by $- 4$ .

$5 + 7 \sqrt{5} i - 4 i \sqrt{5} - 28 i i$

Step 2.1.6.2

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

$5 + 7 \sqrt{5} i - 4 i \sqrt{5} - 28 (i^{1} i)$

Step 2.1.6.3

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

$5 + 7 \sqrt{5} i - 4 i \sqrt{5} - 28 (i^{1} i^{1})$

Step 2.1.6.4

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

$5 + 7 \sqrt{5} i - 4 i \sqrt{5} - 28 i^{1 + 1}$

Step 2.1.6.5

Add $1$ and $1$ .

$5 + 7 \sqrt{5} i - 4 i \sqrt{5} - 28 i^{2}$

Step 2.1.7

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

$5 + 7 \sqrt{5} i - 4 i \sqrt{5} - 28 \cdot - 1$

Step 2.1.8

Multiply $- 28$ by $- 1$ .

$5 + 7 \sqrt{5} i - 4 i \sqrt{5} + 28$

Step 2.2

Add $5$ and $28$ .

$7 \sqrt{5} i - 4 i \sqrt{5} + 33$

Step 2.3

Reorder the factors of $7 \sqrt{5} i$ .

$7 i \sqrt{5} - 4 i \sqrt{5} + 33$

Step 2.4

Subtract $4 i \sqrt{5}$ from $7 i \sqrt{5}$ .

$3 i \sqrt{5} + 33$

Step 3

Reorder $3 i \sqrt{5}$ and $33$ .

$33 + 3 i \sqrt{5}$