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

To write

$- \frac{2}{7}$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 3

To write

$\frac{5}{4}$ as a fraction with a common denominator, multiply by

$\frac{7}{7}$ .

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

Step 4

Write each expression with a common denominator of

$28$ , by multiplying each by an appropriate factor of

$1$ .

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Step 4.1

Multiply

$\frac{2}{7}$ by

$\frac{4}{4}$ .

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

Step 4.2

Multiply

$7$ by

$4$ .

$\frac{5 \sqrt{3}}{6} - \frac{2 \cdot 4}{28} + \frac{5}{4} \cdot \frac{7}{7}$

Step 4.3

Multiply

$\frac{5}{4}$ by

$\frac{7}{7}$ .

$\frac{5 \sqrt{3}}{6} - \frac{2 \cdot 4}{28} + \frac{5 \cdot 7}{4 \cdot 7}$

Step 4.4

Multiply

$4$ by

$7$ .

$\frac{5 \sqrt{3}}{6} - \frac{2 \cdot 4}{28} + \frac{5 \cdot 7}{28}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 \sqrt{3}}{6} + \frac{- 2 \cdot 4 + 5 \cdot 7}{28}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply

$- 2$ by

$4$ .

$\frac{5 \sqrt{3}}{6} + \frac{- 8 + 5 \cdot 7}{28}$

Step 6.2

Multiply

$5$ by

$7$ .

$\frac{5 \sqrt{3}}{6} + \frac{- 8 + 35}{28}$

Step 6.3

Add

$- 8$ and

$35$ .

$\frac{5 \sqrt{3}}{6} + \frac{27}{28}$

Step 7

Factor out the GCF of

$\frac{1}{2}$ from

$\frac{5 \sqrt{3}}{6} + \frac{27}{28}$ .

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Step 7.1

Factor out the GCF of

$\frac{1}{2}$ from each term in the polynomial.

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Step 7.1.1

Factor out the GCF of

$\frac{1}{2}$ from the expression

$\frac{5 \sqrt{3}}{6}$ .

$\frac{1 (\frac{5 \sqrt{3}}{3})}{2} + \frac{27}{28}$

Step 7.1.2

Factor out the GCF of

$\frac{1}{2}$ from the expression

$\frac{27}{28}$ .

$\frac{1 (\frac{5 \sqrt{3}}{3})}{2} + \frac{1 (\frac{27}{14})}{2}$

Step 7.2

Since all the terms share a common factor of

$\frac{1}{2}$ , it can be factored out of each term.

$\frac{1 (\frac{5 \sqrt{3}}{3} + \frac{27}{14})}{2}$

Step 8

The polynomial cannot be factored using the specified method. Try a different method, or if you aren't sure, choose Factor.

The polynomial cannot be factored using the specified method.