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

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

Evaluate

$$\frac{\sqrt{36894726406}}{38416}\approx 5$$

Solution Steps

Calculate 2 to the power of 3 and get 8.

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

Calculate 14 to the power of 8 and get 1475789056.

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

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

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

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

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

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

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Multiply 19208 and 2 to get 38416.

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