Solve sqrt{8!*4/6!+14!/12!*5-582/5}

$$\sqrt { \frac { 8 ! \cdot 4 } { 6 ! } + \frac { 14 ! } { 12 ! \cdot 5 } - \frac { 582 } { 5 } }$$

Evaluate

$$12$$

Solution Steps

The factorial of 8 is 40320.

$$\sqrt{\frac{40320\times 4}{6!}+\frac{14!}{12!\times 5}-\frac{582}{5}}$$

Multiply 40320 and 4 to get 161280.

$$\sqrt{\frac{161280}{6!}+\frac{14!}{12!\times 5}-\frac{582}{5}}$$

The factorial of 6 is 720.

$$\sqrt{\frac{161280}{720}+\frac{14!}{12!\times 5}-\frac{582}{5}}$$

Divide 161280 by 720 to get 224.

$$\sqrt{224+\frac{14!}{12!\times 5}-\frac{582}{5}}$$

The factorial of 14 is 87178291200.

$$\sqrt{224+\frac{87178291200}{12!\times 5}-\frac{582}{5}}$$

The factorial of 12 is 479001600.

$$\sqrt{224+\frac{87178291200}{479001600\times 5}-\frac{582}{5}}$$

Multiply 479001600 and 5 to get 2395008000.

$$\sqrt{224+\frac{87178291200}{2395008000}-\frac{582}{5}}$$

Reduce the fraction \frac{87178291200}{2395008000} to lowest terms by extracting and canceling out 479001600.

$$\sqrt{224+\frac{182}{5}-\frac{582}{5}}$$

Convert 224 to fraction \frac{1120}{5}.

$$\sqrt{\frac{1120}{5}+\frac{182}{5}-\frac{582}{5}}$$

Since \frac{1120}{5} and \frac{182}{5} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1120+182}{5}-\frac{582}{5}}$$

Add 1120 and 182 to get 1302.

$$\sqrt{\frac{1302}{5}-\frac{582}{5}}$$

Since \frac{1302}{5} and \frac{582}{5} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1302-582}{5}}$$

Subtract 582 from 1302 to get 720.

$$\sqrt{\frac{720}{5}}$$

Divide 720 by 5 to get 144.

$$\sqrt{144}$$

Calculate the square root of 144 and get 12.

$$12$$

Factor

$$2^{2}\times 3$$