Basic Math Examples

$(2 \sqrt{2} + 5 \sqrt{3}) (6 \sqrt{2} + 4 \sqrt{3})$

Step 1

Expand $(2 \sqrt{2} + 5 \sqrt{3}) (6 \sqrt{2} + 4 \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$2 \sqrt{2} (6 \sqrt{2} + 4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2} + 4 \sqrt{3})$

Step 1.2

Apply the distributive property.

$2 \sqrt{2} (6 \sqrt{2}) + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2} + 4 \sqrt{3})$

Step 1.3

Apply the distributive property.

$2 \sqrt{2} (6 \sqrt{2}) + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

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 $2 \sqrt{2} (6 \sqrt{2})$ .

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Step 2.1.1.1

Multiply $6$ by $2$ .

$12 \sqrt{2} \sqrt{2} + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.1.2

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

$12 ({\sqrt{2}}^{1} \sqrt{2}) + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.1.3

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

$12 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.1.4

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

$12 {\sqrt{2}}^{1 + 1} + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.1.5

Add $1$ and $1$ .

$12 {\sqrt{2}}^{2} + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

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 $2^{\frac{1}{2}}$ .

$12 {(2^{\frac{1}{2}})}^{2} + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.2.2

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

$12 \cdot 2^{\frac{1}{2} \cdot 2} + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.2.3

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

$12 \cdot 2^{\frac{2}{2}} + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

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.

$12 \cdot 2^{\frac{2}{2}} + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.2.4.2

Rewrite the expression.

$12 \cdot 2^{1} + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.2.5

Evaluate the exponent.

$12 \cdot 2 + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.3

Multiply $12$ by $2$ .

$24 + 2 \sqrt{2} (4 \sqrt{3}) + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.4

Multiply $2 \sqrt{2} (4 \sqrt{3})$ .

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Step 2.1.4.1

Multiply $4$ by $2$ .

$24 + 8 \sqrt{2} \sqrt{3} + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.4.2

Combine using the product rule for radicals.

$24 + 8 \sqrt{3 \cdot 2} + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.4.3

Multiply $3$ by $2$ .

$24 + 8 \sqrt{6} + 5 \sqrt{3} (6 \sqrt{2}) + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.5

Multiply $5 \sqrt{3} (6 \sqrt{2})$ .

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Step 2.1.5.1

Multiply $6$ by $5$ .

$24 + 8 \sqrt{6} + 30 \sqrt{3} \sqrt{2} + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.5.2

Combine using the product rule for radicals.

$24 + 8 \sqrt{6} + 30 \sqrt{2 \cdot 3} + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.5.3

Multiply $2$ by $3$ .

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 5 \sqrt{3} (4 \sqrt{3})$

Step 2.1.6

Multiply $5 \sqrt{3} (4 \sqrt{3})$ .

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Step 2.1.6.1

Multiply $4$ by $5$ .

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 \sqrt{3} \sqrt{3}$

Step 2.1.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.1.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.1.6.4

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

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 {\sqrt{3}}^{1 + 1}$

Step 2.1.6.5

Add $1$ and $1$ .

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 {\sqrt{3}}^{2}$

Step 2.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 {(3^{\frac{1}{2}})}^{2}$

Step 2.1.7.2

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

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 2.1.7.3

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

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 \cdot 3^{\frac{2}{2}}$

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 \cdot 3^{\frac{2}{2}}$

Step 2.1.7.4.2

Rewrite the expression.

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 \cdot 3^{1}$

Step 2.1.7.5

Evaluate the exponent.

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 20 \cdot 3$

Step 2.1.8

Multiply $20$ by $3$ .

$24 + 8 \sqrt{6} + 30 \sqrt{6} + 60$

Step 2.2

Add $24$ and $60$ .

$84 + 8 \sqrt{6} + 30 \sqrt{6}$

Step 2.3

Add $8 \sqrt{6}$ and $30 \sqrt{6}$ .

$84 + 38 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$84 + 38 \sqrt{6}$

Decimal Form:

$177.08061022 \dots$