Basic Math Examples

$((7 \sqrt{3}) \div (\sqrt{10} + \sqrt{3})) - ((2 \sqrt{5}) \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1

Simplify each term.

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Step 1.1

Rewrite the division as a fraction.

$\frac{7 \sqrt{3}}{\sqrt{10} + \sqrt{3}} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.2

Multiply $\frac{7 \sqrt{3}}{\sqrt{10} + \sqrt{3}}$ by $\frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} - \sqrt{3}}$ .

$\frac{7 \sqrt{3}}{\sqrt{10} + \sqrt{3}} \cdot \frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} - \sqrt{3}} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.3

Multiply $\frac{7 \sqrt{3}}{\sqrt{10} + \sqrt{3}}$ by $\frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} - \sqrt{3}}$ .

$\frac{7 \sqrt{3} (\sqrt{10} - \sqrt{3})}{(\sqrt{10} + \sqrt{3}) (\sqrt{10} - \sqrt{3})} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.4

Expand the denominator using the FOIL method.

$\frac{7 \sqrt{3} (\sqrt{10} - \sqrt{3})}{{\sqrt{10}}^{2} - \sqrt{30} + \sqrt{30} - {\sqrt{3}}^{2}} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.5

Simplify.

$\frac{7 \sqrt{3} (\sqrt{10} - \sqrt{3})}{7} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.6

Cancel the common factor of $7$ .

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Step 1.6.1

Cancel the common factor.

$\frac{7 \sqrt{3} (\sqrt{10} - \sqrt{3})}{7} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.6.2

Divide $\sqrt{3} (\sqrt{10} - \sqrt{3})$ by $1$ .

$\sqrt{3} (\sqrt{10} - \sqrt{3}) - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.7

Apply the distributive property.

$\sqrt{3} \sqrt{10} + \sqrt{3} (- \sqrt{3}) - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.8

Combine using the product rule for radicals.

$\sqrt{3 \cdot 10} + \sqrt{3} (- \sqrt{3}) - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.9

Multiply $\sqrt{3} (- \sqrt{3})$ .

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Step 1.9.1

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

$\sqrt{3 \cdot 10} - ({\sqrt{3}}^{1} \sqrt{3}) - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.9.2

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

$\sqrt{3 \cdot 10} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.9.3

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

$\sqrt{3 \cdot 10} - {\sqrt{3}}^{1 + 1} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.9.4

Add $1$ and $1$ .

$\sqrt{3 \cdot 10} - {\sqrt{3}}^{2} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.10

Simplify each term.

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Step 1.10.1

Multiply $3$ by $10$ .

$\sqrt{30} - {\sqrt{3}}^{2} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.10.2

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

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Step 1.10.2.1

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

$\sqrt{30} - {(3^{\frac{1}{2}})}^{2} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.10.2.2

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

$\sqrt{30} - 3^{\frac{1}{2} \cdot 2} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.10.2.3

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

$\sqrt{30} - 3^{\frac{2}{2}} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.10.2.4

Cancel the common factor of $2$ .

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Step 1.10.2.4.1

Cancel the common factor.

$\sqrt{30} - 3^{\frac{2}{2}} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.10.2.4.2

Rewrite the expression.

$\sqrt{30} - 3^{1} - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.10.2.5

Evaluate the exponent.

$\sqrt{30} - 1 \cdot 3 - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.10.3

Multiply $- 1$ by $3$ .

$\sqrt{30} - 3 - (2 \sqrt{5} \div (\sqrt{6} + \sqrt{6})) - (3 \sqrt{2})$

Step 1.11

Rewrite the division as a fraction.

$\sqrt{30} - 3 - \frac{2 \sqrt{5}}{\sqrt{6} + \sqrt{6}} - (3 \sqrt{2})$

Step 1.12

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

$\sqrt{30} - 3 - \frac{2 \sqrt{5}}{2 \sqrt{6}} - (3 \sqrt{2})$

Step 1.13

Cancel the common factor of $2$ .

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Step 1.13.1

Cancel the common factor.

$\sqrt{30} - 3 - \frac{2 \sqrt{5}}{2 \sqrt{6}} - (3 \sqrt{2})$

Step 1.13.2

Rewrite the expression.

$\sqrt{30} - 3 - \frac{\sqrt{5}}{\sqrt{6}} - (3 \sqrt{2})$

Step 1.14

Multiply $\frac{\sqrt{5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{30} - 3 - (\frac{\sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}) - (3 \sqrt{2})$

Step 1.15

Combine and simplify the denominator.

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Step 1.15.1

Multiply $\frac{\sqrt{5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{\sqrt{6} \sqrt{6}} - (3 \sqrt{2})$

Step 1.15.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}} - (3 \sqrt{2})$

Step 1.15.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}} - (3 \sqrt{2})$

Step 1.15.4

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

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1 + 1}} - (3 \sqrt{2})$

Step 1.15.5

Add $1$ and $1$ .

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{2}} - (3 \sqrt{2})$

Step 1.15.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 1.15.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}} - (3 \sqrt{2})$

Step 1.15.6.2

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

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}} - (3 \sqrt{2})$

Step 1.15.6.3

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

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{6^{\frac{2}{2}}} - (3 \sqrt{2})$

Step 1.15.6.4

Cancel the common factor of $2$ .

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Step 1.15.6.4.1

Cancel the common factor.

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{6^{\frac{2}{2}}} - (3 \sqrt{2})$

Step 1.15.6.4.2

Rewrite the expression.

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{6^{1}} - (3 \sqrt{2})$

Step 1.15.6.5

Evaluate the exponent.

$\sqrt{30} - 3 - \frac{\sqrt{5} \sqrt{6}}{6} - (3 \sqrt{2})$

Step 1.16

Simplify the numerator.

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Step 1.16.1

Combine using the product rule for radicals.

$\sqrt{30} - 3 - \frac{\sqrt{5 \cdot 6}}{6} - (3 \sqrt{2})$

Step 1.16.2

Multiply $5$ by $6$ .

$\sqrt{30} - 3 - \frac{\sqrt{30}}{6} - (3 \sqrt{2})$

Step 1.17

Multiply $3$ by $- 1$ .

$\sqrt{30} - 3 - \frac{\sqrt{30}}{6} - 3 \sqrt{2}$

Step 2

Find the common denominator.

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Step 2.1

Write $\sqrt{30}$ as a fraction with denominator $1$ .

$\frac{\sqrt{30}}{1} - 3 - \frac{\sqrt{30}}{6} - 3 \sqrt{2}$

Step 2.2

Multiply $\frac{\sqrt{30}}{1}$ by $\frac{6}{6}$ .

$\frac{\sqrt{30}}{1} \cdot \frac{6}{6} - 3 - \frac{\sqrt{30}}{6} - 3 \sqrt{2}$

Step 2.3

Multiply $\frac{\sqrt{30}}{1}$ by $\frac{6}{6}$ .

$\frac{\sqrt{30} \cdot 6}{6} - 3 - \frac{\sqrt{30}}{6} - 3 \sqrt{2}$

Step 2.4

Write $- 3$ as a fraction with denominator $1$ .

$\frac{\sqrt{30} \cdot 6}{6} + \frac{- 3}{1} - \frac{\sqrt{30}}{6} - 3 \sqrt{2}$

Step 2.5

Multiply $\frac{- 3}{1}$ by $\frac{6}{6}$ .

$\frac{\sqrt{30} \cdot 6}{6} + \frac{- 3}{1} \cdot \frac{6}{6} - \frac{\sqrt{30}}{6} - 3 \sqrt{2}$

Step 2.6

Multiply $\frac{- 3}{1}$ by $\frac{6}{6}$ .

$\frac{\sqrt{30} \cdot 6}{6} + \frac{- 3 \cdot 6}{6} - \frac{\sqrt{30}}{6} - 3 \sqrt{2}$

Step 2.7

Write $- 3 \sqrt{2}$ as a fraction with denominator $1$ .

$\frac{\sqrt{30} \cdot 6}{6} + \frac{- 3 \cdot 6}{6} - \frac{\sqrt{30}}{6} + \frac{- 3 \sqrt{2}}{1}$

Step 2.8

Multiply $\frac{- 3 \sqrt{2}}{1}$ by $\frac{6}{6}$ .

$\frac{\sqrt{30} \cdot 6}{6} + \frac{- 3 \cdot 6}{6} - \frac{\sqrt{30}}{6} + \frac{- 3 \sqrt{2}}{1} \cdot \frac{6}{6}$

Step 2.9

Multiply $\frac{- 3 \sqrt{2}}{1}$ by $\frac{6}{6}$ .

$\frac{\sqrt{30} \cdot 6}{6} + \frac{- 3 \cdot 6}{6} - \frac{\sqrt{30}}{6} + \frac{- 3 \sqrt{2} \cdot 6}{6}$

Step 3

Combine the numerators over the common denominator.

$\frac{\sqrt{30} \cdot 6 - 3 \cdot 6 - \sqrt{30} - 3 \sqrt{2} \cdot 6}{6}$

Step 4

Simplify each term.

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Step 4.1

Move $6$ to the left of $\sqrt{30}$ .

$\frac{6 \cdot \sqrt{30} - 3 \cdot 6 - \sqrt{30} - 3 \sqrt{2} \cdot 6}{6}$

Step 4.2

Multiply $- 3$ by $6$ .

$\frac{6 \sqrt{30} - 18 - \sqrt{30} - 3 \sqrt{2} \cdot 6}{6}$

Step 4.3

Multiply $6$ by $- 3$ .

$\frac{6 \sqrt{30} - 18 - \sqrt{30} - 18 \sqrt{2}}{6}$

Step 5

Simplify terms.

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Step 5.1

Subtract $\sqrt{30}$ from $6 \sqrt{30}$ .

$\frac{- 18 + 5 \sqrt{30} - 18 \sqrt{2}}{6}$

Step 5.2

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{- 1 (18) + 5 \sqrt{30} - 18 \sqrt{2}}{6}$

Step 5.3

Factor $- 1$ out of $5 \sqrt{30}$ .

$\frac{- 1 (18) - (- 5 \sqrt{30}) - 18 \sqrt{2}}{6}$

Step 5.4

Factor $- 1$ out of $- 1 (18) - (- 5 \sqrt{30})$ .

$\frac{- 1 (18 - 5 \sqrt{30}) - 18 \sqrt{2}}{6}$

Step 5.5

Factor $- 1$ out of $- 18 \sqrt{2}$ .

$\frac{- 1 (18 - 5 \sqrt{30}) - (18 \sqrt{2})}{6}$

Step 5.6

Factor $- 1$ out of $- 1 (18 - 5 \sqrt{30}) - (18 \sqrt{2})$ .

$\frac{- 1 (18 - 5 \sqrt{30} + 18 \sqrt{2})}{6}$

Step 5.7

Move the negative in front of the fraction.

$- \frac{18 - 5 \sqrt{30} + 18 \sqrt{2}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{18 - 5 \sqrt{30} + 18 \sqrt{2}}{6}$

Decimal Form:

$- 2.67828604 \dots$