Solve sqrt{(122/521)4^2!}

$$\sqrt{ ( \frac{ 122 }{ 521 } ) { 4 }^{ 2 } ! }$$

Evaluate

$$\frac{241920\sqrt{22723415}}{521}\approx 2213455.782184325$$

Solution Steps

Calculate 4 to the power of 2 and get 16.

$$\sqrt{\frac{122}{521}\times 16!}$$

The factorial of 16 is 20922789888000.

$$\sqrt{\frac{122}{521}\times 20922789888000}$$

Express \frac{122}{521}\times 20922789888000 as a single fraction.

$$\sqrt{\frac{122\times 20922789888000}{521}}$$

Multiply 122 and 20922789888000 to get 2552580366336000.

$$\sqrt{\frac{2552580366336000}{521}}$$

Rewrite the square root of the division \sqrt{\frac{2552580366336000}{521}} as the division of square roots \frac{\sqrt{2552580366336000}}{\sqrt{521}}.

$$\frac{\sqrt{2552580366336000}}{\sqrt{521}}$$

Factor 2552580366336000=241920^{2}\times 43615. Rewrite the square root of the product \sqrt{241920^{2}\times 43615} as the product of square roots \sqrt{241920^{2}}\sqrt{43615}. Take the square root of 241920^{2}.

$$\frac{241920\sqrt{43615}}{\sqrt{521}}$$

Rationalize the denominator of \frac{241920\sqrt{43615}}{\sqrt{521}} by multiplying numerator and denominator by \sqrt{521}.

$$\frac{241920\sqrt{43615}\sqrt{521}}{\left(\sqrt{521}\right)^{2}}$$

The square of \sqrt{521} is 521.

$$\frac{241920\sqrt{43615}\sqrt{521}}{521}$$

To multiply \sqrt{43615} and \sqrt{521}, multiply the numbers under the square root.

$$\frac{241920\sqrt{22723415}}{521}$$