Basic Math Examples

$(\sqrt{6} - \sqrt{5 i}) (\sqrt{6} + \sqrt{5 i})$

Step 1

Expand $(\sqrt{6} - \sqrt{5 i}) (\sqrt{6} + \sqrt{5 i})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{6} (\sqrt{6} + \sqrt{5 i}) - \sqrt{5 i} (\sqrt{6} + \sqrt{5 i})$

Step 1.2

Apply the distributive property.

$\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{5 i} - \sqrt{5 i} (\sqrt{6} + \sqrt{5 i})$

Step 1.3

Apply the distributive property.

$\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{5 i} - \sqrt{5 i} \sqrt{6} - \sqrt{5 i} \sqrt{5 i}$

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.

$\sqrt{6 \cdot 6} + \sqrt{6} \sqrt{5 i} - \sqrt{5 i} \sqrt{6} - \sqrt{5 i} \sqrt{5 i}$

Step 2.1.2

Multiply $6$ by $6$ .

$\sqrt{36} + \sqrt{6} \sqrt{5 i} - \sqrt{5 i} \sqrt{6} - \sqrt{5 i} \sqrt{5 i}$

Step 2.1.3

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2}} + \sqrt{6} \sqrt{5 i} - \sqrt{5 i} \sqrt{6} - \sqrt{5 i} \sqrt{5 i}$

Step 2.1.4

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

$6 + \sqrt{6} \sqrt{5 i} - \sqrt{5 i} \sqrt{6} - \sqrt{5 i} \sqrt{5 i}$

Step 2.1.5

Combine using the product rule for radicals.

$6 + \sqrt{6 (5 i)} - \sqrt{5 i} \sqrt{6} - \sqrt{5 i} \sqrt{5 i}$

Step 2.1.6

Multiply $6$ by $5$ .

$6 + \sqrt{30 i} - \sqrt{5 i} \sqrt{6} - \sqrt{5 i} \sqrt{5 i}$

Step 2.1.7

Multiply $- \sqrt{5 i} \sqrt{6}$ .

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Step 2.1.7.1

Combine using the product rule for radicals.

$6 + \sqrt{30 i} - \sqrt{6 (5 i)} - \sqrt{5 i} \sqrt{5 i}$

Step 2.1.7.2

Multiply $6$ by $5$ .

$6 + \sqrt{30 i} - \sqrt{30 i} - \sqrt{5 i} \sqrt{5 i}$

Step 2.1.8

Multiply $- \sqrt{5 i} \sqrt{5 i}$ .

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Step 2.1.8.1

Raise $\sqrt{5 i}$ to the power of $1$ .

$6 + \sqrt{30 i} - \sqrt{30 i} - ({\sqrt{5 i}}^{1} \sqrt{5 i})$

Step 2.1.8.2

Raise $\sqrt{5 i}$ to the power of $1$ .

$6 + \sqrt{30 i} - \sqrt{30 i} - ({\sqrt{5 i}}^{1} {\sqrt{5 i}}^{1})$

Step 2.1.8.3

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

$6 + \sqrt{30 i} - \sqrt{30 i} - {\sqrt{5 i}}^{1 + 1}$

Step 2.1.8.4

Add $1$ and $1$ .

$6 + \sqrt{30 i} - \sqrt{30 i} - {\sqrt{5 i}}^{2}$

Step 2.1.9

Rewrite ${\sqrt{5 i}}^{2}$ as $5 i$ .

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Step 2.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 i}$ as ${(5 i)}^{\frac{1}{2}}$ .

$6 + \sqrt{30 i} - \sqrt{30 i} - {({(5 i)}^{\frac{1}{2}})}^{2}$

Step 2.1.9.2

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

$6 + \sqrt{30 i} - \sqrt{30 i} - {(5 i)}^{\frac{1}{2} \cdot 2}$

Step 2.1.9.3

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

$6 + \sqrt{30 i} - \sqrt{30 i} - {(5 i)}^{\frac{2}{2}}$

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

$6 + \sqrt{30 i} - \sqrt{30 i} - {(5 i)}^{\frac{2}{2}}$

Step 2.1.9.4.2

Rewrite the expression.

$6 + \sqrt{30 i} - \sqrt{30 i} - {(5 i)}^{1}$

Step 2.1.9.5

Simplify.

$6 + \sqrt{30 i} - \sqrt{30 i} - (5 i)$

Step 2.1.10

Multiply $5$ by $- 1$ .

$6 + \sqrt{30 i} - \sqrt{30 i} - 5 i$

Step 2.2

Subtract $\sqrt{30 i}$ from $\sqrt{30 i}$ .

$6 + 0 - 5 i$

Step 2.3

Add $6$ and $0$ .

$6 - 5 i$