Finite Math Examples

$\frac{7}{3 \sqrt{2}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{7}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{7}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{7}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{7 \sqrt{2}}{3 \sqrt{2} \sqrt{2}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.2

Move $\sqrt{2}$ .

$\frac{7 \sqrt{2}}{3 (\sqrt{2} \sqrt{2})} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.3

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

$\frac{7 \sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.4

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

$\frac{7 \sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.5

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

$\frac{7 \sqrt{2}}{3 {\sqrt{2}}^{1 + 1}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.6

Add $1$ and $1$ .

$\frac{7 \sqrt{2}}{3 {\sqrt{2}}^{2}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.7

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

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Step 1.2.7.1

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

$\frac{7 \sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.7.2

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

$\frac{7 \sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.7.3

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

$\frac{7 \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

$\frac{7 \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.7.4.2

Rewrite the expression.

$\frac{7 \sqrt{2}}{3 \cdot 2^{1}} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.2.7.5

Evaluate the exponent.

$\frac{7 \sqrt{2}}{3 \cdot 2} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.3

Multiply $3$ by $2$ .

$\frac{7 \sqrt{2}}{6} + \frac{3}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.4

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{7 \sqrt{2}}{6} + \frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

Multiply $\frac{3}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{\sqrt{2} \sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.5.2

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

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} - \frac{\sqrt{12}}{3}$

Step 1.5.3

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

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} - \frac{\sqrt{12}}{3}$

Step 1.5.4

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

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} - \frac{\sqrt{12}}{3}$

Step 1.5.5

Add $1$ and $1$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{{\sqrt{2}}^{2}} - \frac{\sqrt{12}}{3}$

Step 1.5.6

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

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Step 1.5.6.1

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

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} - \frac{\sqrt{12}}{3}$

Step 1.5.6.2

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

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} - \frac{\sqrt{12}}{3}$

Step 1.5.6.3

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

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2^{\frac{2}{2}}} - \frac{\sqrt{12}}{3}$

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2^{\frac{2}{2}}} - \frac{\sqrt{12}}{3}$

Step 1.5.6.4.2

Rewrite the expression.

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2^{1}} - \frac{\sqrt{12}}{3}$

Step 1.5.6.5

Evaluate the exponent.

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2} - \frac{\sqrt{12}}{3}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.6.1.1

Factor $4$ out of $12$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2} - \frac{\sqrt{4 (3)}}{3}$

Step 1.6.1.2

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

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2} - \frac{\sqrt{2^{2} \cdot 3}}{3}$

Step 1.6.2

Pull terms out from under the radical.

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2} - \frac{2 \sqrt{3}}{3}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{3}{3}$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2}}{2} \cdot \frac{3}{3} - \frac{2 \sqrt{3}}{3}$

Step 2.2

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{3}{3}$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2} \cdot 3}{2 \cdot 3} - \frac{2 \sqrt{3}}{3}$

Step 2.3

Multiply $\frac{2 \sqrt{3}}{3}$ by $\frac{2}{2}$ .

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

Step 2.4

Multiply $\frac{2 \sqrt{3}}{3}$ by $\frac{2}{2}$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2} \cdot 3}{2 \cdot 3} - \frac{2 \sqrt{3} \cdot 2}{3 \cdot 2}$

Step 2.5

Multiply $2$ by $3$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2} \cdot 3}{6} - \frac{2 \sqrt{3} \cdot 2}{3 \cdot 2}$

Step 2.6

Reorder the factors of $3 \cdot 2$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2} \cdot 3}{6} - \frac{2 \sqrt{3} \cdot 2}{2 \cdot 3}$

Step 2.7

Multiply $2$ by $3$ .

$\frac{7 \sqrt{2}}{6} + \frac{3 \sqrt{2} \cdot 3}{6} - \frac{2 \sqrt{3} \cdot 2}{6}$

Step 3

Combine the numerators over the common denominator.

$\frac{7 \sqrt{2} + 3 \sqrt{2} \cdot 3 - 2 \sqrt{3} \cdot 2}{6}$

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $3$ .

$\frac{7 \sqrt{2} + 9 \sqrt{2} - 2 \sqrt{3} \cdot 2}{6}$

Step 4.2

Multiply $2$ by $- 2$ .

$\frac{7 \sqrt{2} + 9 \sqrt{2} - 4 \sqrt{3}}{6}$

Step 5

Simplify terms.

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Step 5.1

Add $7 \sqrt{2}$ and $9 \sqrt{2}$ .

$\frac{16 \sqrt{2} - 4 \sqrt{3}}{6}$

Step 5.2

Cancel the common factor of $16 \sqrt{2} - 4 \sqrt{3}$ and $6$ .

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Step 5.2.1

Factor $2$ out of $16 \sqrt{2}$ .

$\frac{2 (8 \sqrt{2}) - 4 \sqrt{3}}{6}$

Step 5.2.2

Factor $2$ out of $- 4 \sqrt{3}$ .

$\frac{2 (8 \sqrt{2}) + 2 (- 2 \sqrt{3})}{6}$

Step 5.2.3

Factor $2$ out of $2 (8 \sqrt{2}) + 2 (- 2 \sqrt{3})$ .

$\frac{2 (8 \sqrt{2} - 2 \sqrt{3})}{6}$

Step 5.2.4

Cancel the common factors.

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Step 5.2.4.1

Factor $2$ out of $6$ .

$\frac{2 (8 \sqrt{2} - 2 \sqrt{3})}{2 (3)}$

Step 5.2.4.2

Cancel the common factor.

$\frac{2 (8 \sqrt{2} - 2 \sqrt{3})}{2 \cdot 3}$

Step 5.2.4.3

Rewrite the expression.

$\frac{8 \sqrt{2} - 2 \sqrt{3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{8 \sqrt{2} - 2 \sqrt{3}}{3}$

Decimal Form:

$2.61653562 \dots$