Basic Math Examples

$(7 \sqrt{3} - \sqrt{2}) (\sqrt{3} + 5 \sqrt{2})$

Step 1

Expand $(7 \sqrt{3} - \sqrt{2}) (\sqrt{3} + 5 \sqrt{2})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$7 \sqrt{3} (\sqrt{3} + 5 \sqrt{2}) - \sqrt{2} (\sqrt{3} + 5 \sqrt{2})$

Step 1.2

Apply the distributive property.

$7 \sqrt{3} \sqrt{3} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} (\sqrt{3} + 5 \sqrt{2})$

Step 1.3

Apply the distributive property.

$7 \sqrt{3} \sqrt{3} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

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 $7 \sqrt{3} \sqrt{3}$ .

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Step 2.1.1.1

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

$7 ({\sqrt{3}}^{1} \sqrt{3}) + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.1.2

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

$7 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.1.3

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

$7 {\sqrt{3}}^{1 + 1} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.1.4

Add $1$ and $1$ .

$7 {\sqrt{3}}^{2} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.2

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

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Step 2.1.2.1

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

$7 {(3^{\frac{1}{2}})}^{2} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.2.2

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

$7 \cdot 3^{\frac{1}{2} \cdot 2} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.2.3

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

$7 \cdot 3^{\frac{2}{2}} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

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.

$7 \cdot 3^{\frac{2}{2}} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.2.4.2

Rewrite the expression.

$7 \cdot 3^{1} + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.2.5

Evaluate the exponent.

$7 \cdot 3 + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.3

Multiply $7$ by $3$ .

$21 + 7 \sqrt{3} (5 \sqrt{2}) - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.4

Multiply $7 \sqrt{3} (5 \sqrt{2})$ .

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Step 2.1.4.1

Multiply $5$ by $7$ .

$21 + 35 \sqrt{3} \sqrt{2} - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.4.2

Combine using the product rule for radicals.

$21 + 35 \sqrt{2 \cdot 3} - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.4.3

Multiply $2$ by $3$ .

$21 + 35 \sqrt{6} - \sqrt{2} \sqrt{3} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.5

Multiply $- \sqrt{2} \sqrt{3}$ .

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Step 2.1.5.1

Combine using the product rule for radicals.

$21 + 35 \sqrt{6} - \sqrt{3 \cdot 2} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.5.2

Multiply $3$ by $2$ .

$21 + 35 \sqrt{6} - \sqrt{6} - \sqrt{2} (5 \sqrt{2})$

Step 2.1.6

Multiply $- \sqrt{2} (5 \sqrt{2})$ .

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Step 2.1.6.1

Multiply $5$ by $- 1$ .

$21 + 35 \sqrt{6} - \sqrt{6} - 5 \sqrt{2} \sqrt{2}$

Step 2.1.6.2

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

$21 + 35 \sqrt{6} - \sqrt{6} - 5 ({\sqrt{2}}^{1} \sqrt{2})$

Step 2.1.6.3

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

$21 + 35 \sqrt{6} - \sqrt{6} - 5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 2.1.6.4

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

$21 + 35 \sqrt{6} - \sqrt{6} - 5 {\sqrt{2}}^{1 + 1}$

Step 2.1.6.5

Add $1$ and $1$ .

$21 + 35 \sqrt{6} - \sqrt{6} - 5 {\sqrt{2}}^{2}$

Step 2.1.7

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

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Step 2.1.7.1

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

$21 + 35 \sqrt{6} - \sqrt{6} - 5 {(2^{\frac{1}{2}})}^{2}$

Step 2.1.7.2

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

$21 + 35 \sqrt{6} - \sqrt{6} - 5 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 2.1.7.3

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

$21 + 35 \sqrt{6} - \sqrt{6} - 5 \cdot 2^{\frac{2}{2}}$

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.

$21 + 35 \sqrt{6} - \sqrt{6} - 5 \cdot 2^{\frac{2}{2}}$

Step 2.1.7.4.2

Rewrite the expression.

$21 + 35 \sqrt{6} - \sqrt{6} - 5 \cdot 2^{1}$

Step 2.1.7.5

Evaluate the exponent.

$21 + 35 \sqrt{6} - \sqrt{6} - 5 \cdot 2$

Step 2.1.8

Multiply $- 5$ by $2$ .

$21 + 35 \sqrt{6} - \sqrt{6} - 10$

Step 2.2

Subtract $10$ from $21$ .

$11 + 35 \sqrt{6} - \sqrt{6}$

Step 2.3

Subtract $\sqrt{6}$ from $35 \sqrt{6}$ .

$11 + 34 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$11 + 34 \sqrt{6}$

Decimal Form:

$94.28265125 \dots$