Basic Math Examples

$(\sqrt{2} + 9 \sqrt{10}) (2 \sqrt{2} - 6 \sqrt{10})$

Step 1

Expand $(\sqrt{2} + 9 \sqrt{10}) (2 \sqrt{2} - 6 \sqrt{10})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{2} (2 \sqrt{2} - 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2} - 6 \sqrt{10})$

Step 1.2

Apply the distributive property.

$\sqrt{2} (2 \sqrt{2}) + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2} - 6 \sqrt{10})$

Step 1.3

Apply the distributive property.

$\sqrt{2} (2 \sqrt{2}) + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

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

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

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Step 2.1.1.1

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

$2 ({\sqrt{2}}^{1} \sqrt{2}) + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.1.2

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

$2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.1.3

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

$2 {\sqrt{2}}^{1 + 1} + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.1.4

Add $1$ and $1$ .

$2 {\sqrt{2}}^{2} + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.2

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

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Step 2.1.2.1

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

$2 {(2^{\frac{1}{2}})}^{2} + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.2.2

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

$2 \cdot 2^{\frac{1}{2} \cdot 2} + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.2.3

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

$2 \cdot 2^{\frac{2}{2}} + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

$2 \cdot 2^{\frac{2}{2}} + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.2.4.2

Rewrite the expression.

$2 \cdot 2^{1} + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.2.5

Evaluate the exponent.

$2 \cdot 2 + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.3

Multiply $2$ by $2$ .

$4 + \sqrt{2} (- 6 \sqrt{10}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.4

Multiply $\sqrt{2} (- 6 \sqrt{10})$ .

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Step 2.1.4.1

Combine using the product rule for radicals.

$4 - 6 \sqrt{2 \cdot 10} + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.4.2

Multiply $2$ by $10$ .

$4 - 6 \sqrt{20} + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.5

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.1.5.1

Factor $4$ out of $20$ .

$4 - 6 \sqrt{4 (5)} + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.5.2

Rewrite $4$ as $2^{2}$ .

$4 - 6 \sqrt{2^{2} \cdot 5} + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.6

Pull terms out from under the radical.

$4 - 6 (2 \sqrt{5}) + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.7

Multiply $2$ by $- 6$ .

$4 - 12 \sqrt{5} + 9 \sqrt{10} (2 \sqrt{2}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.8

Multiply $9 \sqrt{10} (2 \sqrt{2})$ .

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Step 2.1.8.1

Multiply $2$ by $9$ .

$4 - 12 \sqrt{5} + 18 \sqrt{10} \sqrt{2} + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.8.2

Combine using the product rule for radicals.

$4 - 12 \sqrt{5} + 18 \sqrt{2 \cdot 10} + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.8.3

Multiply $2$ by $10$ .

$4 - 12 \sqrt{5} + 18 \sqrt{20} + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.9

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.1.9.1

Factor $4$ out of $20$ .

$4 - 12 \sqrt{5} + 18 \sqrt{4 (5)} + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.9.2

Rewrite $4$ as $2^{2}$ .

$4 - 12 \sqrt{5} + 18 \sqrt{2^{2} \cdot 5} + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.10

Pull terms out from under the radical.

$4 - 12 \sqrt{5} + 18 (2 \sqrt{5}) + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.11

Multiply $2$ by $18$ .

$4 - 12 \sqrt{5} + 36 \sqrt{5} + 9 \sqrt{10} (- 6 \sqrt{10})$

Step 2.1.12

Multiply $9 \sqrt{10} (- 6 \sqrt{10})$ .

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Step 2.1.12.1

Multiply $- 6$ by $9$ .

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 \sqrt{10} \sqrt{10}$

Step 2.1.12.2

Raise $\sqrt{10}$ to the power of $1$ .

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 ({\sqrt{10}}^{1} \sqrt{10})$

Step 2.1.12.3

Raise $\sqrt{10}$ to the power of $1$ .

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})$

Step 2.1.12.4

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

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 {\sqrt{10}}^{1 + 1}$

Step 2.1.12.5

Add $1$ and $1$ .

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 {\sqrt{10}}^{2}$

Step 2.1.13

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.1.13.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 {(10^{\frac{1}{2}})}^{2}$

Step 2.1.13.2

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

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 \cdot 10^{\frac{1}{2} \cdot 2}$

Step 2.1.13.3

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

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 \cdot 10^{\frac{2}{2}}$

Step 2.1.13.4

Cancel the common factor of $2$ .

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Step 2.1.13.4.1

Cancel the common factor.

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 \cdot 10^{\frac{2}{2}}$

Step 2.1.13.4.2

Rewrite the expression.

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 \cdot 10^{1}$

Step 2.1.13.5

Evaluate the exponent.

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 54 \cdot 10$

Step 2.1.14

Multiply $- 54$ by $10$ .

$4 - 12 \sqrt{5} + 36 \sqrt{5} - 540$

Step 2.2

Subtract $540$ from $4$ .

$- 536 - 12 \sqrt{5} + 36 \sqrt{5}$

Step 2.3

Add $- 12 \sqrt{5}$ and $36 \sqrt{5}$ .

$- 536 + 24 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 536 + 24 \sqrt{5}$

Decimal Form:

$- 482.33436854 \dots$