Finite Math Examples

$\sqrt{\frac{3}{4} \div {(1 - \frac{2}{5})}^{2}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1

Simplify each term.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{3}{4} \div {(\frac{5}{5} - \frac{2}{5})}^{2}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.2

Combine the numerators over the common denominator.

$\sqrt{\frac{3}{4} \div {(\frac{5 - 2}{5})}^{2}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.3

Subtract $2$ from $5$ .

$\sqrt{\frac{3}{4} \div {(\frac{3}{5})}^{2}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.4

Apply the product rule to $\frac{3}{5}$ .

$\sqrt{\frac{3}{4} \div \frac{3^{2}}{5^{2}}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.5

To divide by a fraction, multiply by its reciprocal.

$\sqrt{\frac{3}{4} \cdot \frac{5^{2}}{3^{2}}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.6

Combine.

$\sqrt{\frac{3 \cdot 5^{2}}{4 \cdot 3^{2}}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.7

Cancel the common factor of $3$ and $3^{2}$ .

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Step 1.7.1

Factor $3$ out of $3 \cdot 5^{2}$ .

$\sqrt{\frac{3 (5^{2})}{4 \cdot 3^{2}}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $3$ out of $4 \cdot 3^{2}$ .

$\sqrt{\frac{3 (5^{2})}{3 (4 \cdot 3)}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.7.2.2

Cancel the common factor.

$\sqrt{\frac{3 \cdot 5^{2}}{3 (4 \cdot 3)}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.7.2.3

Rewrite the expression.

$\sqrt{\frac{5^{2}}{4 \cdot 3}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.8

Raise $5$ to the power of $2$ .

$\sqrt{\frac{25}{4 \cdot 3}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.9

Multiply $4$ by $3$ .

$\sqrt{\frac{25}{12}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.10

Rewrite $\sqrt{\frac{25}{12}}$ as $\frac{\sqrt{25}}{\sqrt{12}}$ .

$\frac{\sqrt{25}}{\sqrt{12}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.11

Simplify the numerator.

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Step 1.11.1

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{12}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.11.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{5}{\sqrt{12}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.12

Simplify the denominator.

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Step 1.12.1

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

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Step 1.12.1.1

Factor $4$ out of $12$ .

$\frac{5}{\sqrt{4 (3)}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.12.1.2

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

$\frac{5}{\sqrt{2^{2} \cdot 3}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.12.2

Pull terms out from under the radical.

$\frac{5}{2 \sqrt{3}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.13

Multiply $\frac{5}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{5}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14

Combine and simplify the denominator.

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Step 1.14.1

Multiply $\frac{5}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{5 \sqrt{3}}{2 \sqrt{3} \sqrt{3}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.2

Move $\sqrt{3}$ .

$\frac{5 \sqrt{3}}{2 (\sqrt{3} \sqrt{3})} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.3

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

$\frac{5 \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.4

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

$\frac{5 \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.5

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

$\frac{5 \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.6

Add $1$ and $1$ .

$\frac{5 \sqrt{3}}{2 {\sqrt{3}}^{2}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.7

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

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Step 1.14.7.1

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

$\frac{5 \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.7.2

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

$\frac{5 \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.7.3

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

$\frac{5 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.7.4

Cancel the common factor of $2$ .

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Step 1.14.7.4.1

Cancel the common factor.

$\frac{5 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.7.4.2

Rewrite the expression.

$\frac{5 \sqrt{3}}{2 \cdot 3^{1}} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.14.7.5

Evaluate the exponent.

$\frac{5 \sqrt{3}}{2 \cdot 3} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.15

Multiply $2$ by $3$ .

$\frac{5 \sqrt{3}}{6} - \frac{4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.16

Cancel the common factor of $2$ .

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Step 1.16.1

Move the leading negative in $- \frac{4}{7}$ into the numerator.

$\frac{5 \sqrt{3}}{6} + \frac{- 4}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.16.2

Factor $2$ out of $- 4$ .

$\frac{5 \sqrt{3}}{6} + \frac{2 (- 2)}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.16.3

Cancel the common factor.

$\frac{5 \sqrt{3}}{6} + \frac{2 \cdot - 2}{7} \cdot \frac{1}{2} + \frac{1}{4} \cdot 5$

Step 1.16.4

Rewrite the expression.

$\frac{5 \sqrt{3}}{6} + \frac{- 2}{7} + \frac{1}{4} \cdot 5$

Step 1.17

Move the negative in front of the fraction.

$\frac{5 \sqrt{3}}{6} - \frac{2}{7} + \frac{1}{4} \cdot 5$

Step 1.18

Combine $\frac{1}{4}$ and $5$ .

$\frac{5 \sqrt{3}}{6} - \frac{2}{7} + \frac{5}{4}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{5 \sqrt{3}}{6}$ by $\frac{14}{14}$ .

$\frac{5 \sqrt{3}}{6} \cdot \frac{14}{14} - \frac{2}{7} + \frac{5}{4}$

Step 2.2

Multiply $\frac{5 \sqrt{3}}{6}$ by $\frac{14}{14}$ .

$\frac{5 \sqrt{3} \cdot 14}{6 \cdot 14} - \frac{2}{7} + \frac{5}{4}$

Step 2.3

Multiply $\frac{2}{7}$ by $\frac{12}{12}$ .

$\frac{5 \sqrt{3} \cdot 14}{6 \cdot 14} - (\frac{2}{7} \cdot \frac{12}{12}) + \frac{5}{4}$

Step 2.4

Multiply $\frac{2}{7}$ by $\frac{12}{12}$ .

$\frac{5 \sqrt{3} \cdot 14}{6 \cdot 14} - \frac{2 \cdot 12}{7 \cdot 12} + \frac{5}{4}$

Step 2.5

Multiply $\frac{5}{4}$ by $\frac{21}{21}$ .

$\frac{5 \sqrt{3} \cdot 14}{6 \cdot 14} - \frac{2 \cdot 12}{7 \cdot 12} + \frac{5}{4} \cdot \frac{21}{21}$

Step 2.6

Multiply $\frac{5}{4}$ by $\frac{21}{21}$ .

$\frac{5 \sqrt{3} \cdot 14}{6 \cdot 14} - \frac{2 \cdot 12}{7 \cdot 12} + \frac{5 \cdot 21}{4 \cdot 21}$

Step 2.7

Multiply $6$ by $14$ .

$\frac{5 \sqrt{3} \cdot 14}{84} - \frac{2 \cdot 12}{7 \cdot 12} + \frac{5 \cdot 21}{4 \cdot 21}$

Step 2.8

Multiply $7$ by $12$ .

$\frac{5 \sqrt{3} \cdot 14}{84} - \frac{2 \cdot 12}{84} + \frac{5 \cdot 21}{4 \cdot 21}$

Step 2.9

Multiply $4$ by $21$ .

$\frac{5 \sqrt{3} \cdot 14}{84} - \frac{2 \cdot 12}{84} + \frac{5 \cdot 21}{84}$

Step 3

Combine the numerators over the common denominator.

$\frac{5 \sqrt{3} \cdot 14 - 2 \cdot 12 + 5 \cdot 21}{84}$

Step 4

Simplify each term.

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Step 4.1

Multiply $14$ by $5$ .

$\frac{70 \sqrt{3} - 2 \cdot 12 + 5 \cdot 21}{84}$

Step 4.2

Multiply $- 2$ by $12$ .

$\frac{70 \sqrt{3} - 24 + 5 \cdot 21}{84}$

Step 4.3

Multiply $5$ by $21$ .

$\frac{70 \sqrt{3} - 24 + 105}{84}$

Step 5

Add $- 24$ and $105$ .

$\frac{70 \sqrt{3} + 81}{84}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{70 \sqrt{3} + 81}{84}$

Decimal Form:

$2.40766138 \dots$