Pre-Algebra Examples

$(\sqrt{5} + 9 i) (\sqrt{5} - 9 i)$

Step 1

Expand $(\sqrt{5} + 9 i) (\sqrt{5} - 9 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{5} (\sqrt{5} - 9 i) + 9 i (\sqrt{5} - 9 i)$

Step 1.2

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} (- 9 i) + 9 i (\sqrt{5} - 9 i)$

Step 1.3

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} (- 9 i) + 9 i \sqrt{5} + 9 i (- 9 i)$

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.

$\sqrt{5 \cdot 5} + \sqrt{5} (- 9 i) + 9 i \sqrt{5} + 9 i (- 9 i)$

Step 2.1.2

Multiply $5$ by $5$ .

$\sqrt{25} + \sqrt{5} (- 9 i) + 9 i \sqrt{5} + 9 i (- 9 i)$

Step 2.1.3

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

$\sqrt{5^{2}} + \sqrt{5} (- 9 i) + 9 i \sqrt{5} + 9 i (- 9 i)$

Step 2.1.4

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

$5 + \sqrt{5} (- 9 i) + 9 i \sqrt{5} + 9 i (- 9 i)$

Step 2.1.5

Multiply $9 i (- 9 i)$ .

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Step 2.1.5.1

Multiply $- 9$ by $9$ .

$5 + \sqrt{5} (- 9 i) + 9 i \sqrt{5} - 81 i i$

Step 2.1.5.2

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

$5 + \sqrt{5} (- 9 i) + 9 i \sqrt{5} - 81 (i^{1} i)$

Step 2.1.5.3

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

$5 + \sqrt{5} (- 9 i) + 9 i \sqrt{5} - 81 (i^{1} i^{1})$

Step 2.1.5.4

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

$5 + \sqrt{5} (- 9 i) + 9 i \sqrt{5} - 81 i^{1 + 1}$

Step 2.1.5.5

Add $1$ and $1$ .

$5 + \sqrt{5} (- 9 i) + 9 i \sqrt{5} - 81 i^{2}$

Step 2.1.6

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

$5 + \sqrt{5} (- 9 i) + 9 i \sqrt{5} - 81 \cdot - 1$

Step 2.1.7

Multiply $- 81$ by $- 1$ .

$5 + \sqrt{5} (- 9 i) + 9 i \sqrt{5} + 81$

Step 2.2

Add $5$ and $81$ .

$\sqrt{5} (- 9 i) + 9 i \sqrt{5} + 86$

Step 2.3

Reorder the factors of $\sqrt{5} (- 9 i)$ .

$- 9 i \sqrt{5} + 9 i \sqrt{5} + 86$

Step 2.4

Add $- 9 i \sqrt{5}$ and $9 i \sqrt{5}$ .

$0 + 86$

Step 2.5

Add $0$ and $86$ .

$86$