Solve 748*25+sqrt{25+6div14^2^3}

$$748 \times 25+ \sqrt{ 25+6 \div { 14 }^{ { 2 }^{ 3 } } }$$

Evaluate

$$\frac{\sqrt{36894726406}}{38416}+18700\approx 18705$$

Short Solution Steps

Multiply 748 and 25 to get 18700.

$$18700+\sqrt{25+\frac{6}{14^{2^{3}}}}$$

Calculate 2 to the power of 3 and get 8.

$$18700+\sqrt{25+\frac{6}{14^{8}}}$$

Calculate 14 to the power of 8 and get 1475789056.

$$18700+\sqrt{25+\frac{6}{1475789056}}$$

Reduce the fraction \frac{6}{1475789056} to lowest terms by extracting and canceling out 2.

$$18700+\sqrt{25+\frac{3}{737894528}}$$

Add 25 and \frac{3}{737894528} to get \frac{18447363203}{737894528}.

$$18700+\sqrt{\frac{18447363203}{737894528}}$$

Rewrite the square root of the division \sqrt{\frac{18447363203}{737894528}} as the division of square roots \frac{\sqrt{18447363203}}{\sqrt{737894528}}.

$$18700+\frac{\sqrt{18447363203}}{\sqrt{737894528}}$$

Factor 737894528=19208^{2}\times 2. Rewrite the square root of the product \sqrt{19208^{2}\times 2} as the product of square roots \sqrt{19208^{2}}\sqrt{2}. Take the square root of 19208^{2}.

$$18700+\frac{\sqrt{18447363203}}{19208\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{18447363203}}{19208\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$18700+\frac{\sqrt{18447363203}\sqrt{2}}{19208\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$18700+\frac{\sqrt{18447363203}\sqrt{2}}{19208\times 2}$$

To multiply \sqrt{18447363203} and \sqrt{2}, multiply the numbers under the square root.

$$18700+\frac{\sqrt{36894726406}}{19208\times 2}$$

Multiply 19208 and 2 to get 38416.

$$18700+\frac{\sqrt{36894726406}}{38416}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 18700 times \frac{38416}{38416}.

$$\frac{18700\times 38416}{38416}+\frac{\sqrt{36894726406}}{38416}$$

Since \frac{18700\times 38416}{38416} and \frac{\sqrt{36894726406}}{38416} have the same denominator, add them by adding their numerators.

$$\frac{18700\times 38416+\sqrt{36894726406}}{38416}$$

Do the multiplications in 18700\times 38416+\sqrt{36894726406}.

$$\frac{718379200+\sqrt{36894726406}}{38416}$$