Solve 5/sqrt{1frac{1{36}}*sqrt{112/37}+frac{sqrt{75}}{sqrt{3}}}

$$\frac { 5 } { \sqrt { 1 \frac { 1 } { 36 } } \cdot \sqrt { 1 \frac { 12 } { 37 } } + \frac { \sqrt { 75 } } { \sqrt { 3 } } }$$

Evaluate

$$\frac{30}{37}\approx 0.810810811$$

Short Solution Steps

Multiply 1 and 36 to get 36.

$$\frac{5}{\sqrt{\frac{36+1}{36}}\sqrt{\frac{1\times 37+12}{37}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Add 36 and 1 to get 37.

$$\frac{5}{\sqrt{\frac{37}{36}}\sqrt{\frac{1\times 37+12}{37}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Rewrite the square root of the division \sqrt{\frac{37}{36}} as the division of square roots \frac{\sqrt{37}}{\sqrt{36}}.

$$\frac{5}{\frac{\sqrt{37}}{\sqrt{36}}\sqrt{\frac{1\times 37+12}{37}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Calculate the square root of 36 and get 6.

$$\frac{5}{\frac{\sqrt{37}}{6}\sqrt{\frac{1\times 37+12}{37}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Multiply 1 and 37 to get 37.

$$\frac{5}{\frac{\sqrt{37}}{6}\sqrt{\frac{37+12}{37}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Add 37 and 12 to get 49.

$$\frac{5}{\frac{\sqrt{37}}{6}\sqrt{\frac{49}{37}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Rewrite the square root of the division \sqrt{\frac{49}{37}} as the division of square roots \frac{\sqrt{49}}{\sqrt{37}}.

$$\frac{5}{\frac{\sqrt{37}}{6}\times \frac{\sqrt{49}}{\sqrt{37}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Calculate the square root of 49 and get 7.

$$\frac{5}{\frac{\sqrt{37}}{6}\times \frac{7}{\sqrt{37}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Rationalize the denominator of \frac{7}{\sqrt{37}} by multiplying numerator and denominator by \sqrt{37}.

$$\frac{5}{\frac{\sqrt{37}}{6}\times \frac{7\sqrt{37}}{\left(\sqrt{37}\right)^{2}}+\frac{\sqrt{75}}{\sqrt{3}}}$$

The square of \sqrt{37} is 37.

$$\frac{5}{\frac{\sqrt{37}}{6}\times \frac{7\sqrt{37}}{37}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Multiply \frac{\sqrt{37}}{6} times \frac{7\sqrt{37}}{37} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5}{\frac{\sqrt{37}\times 7\sqrt{37}}{6\times 37}+\frac{\sqrt{75}}{\sqrt{3}}}$$

Rewrite the division of square roots \frac{\sqrt{75}}{\sqrt{3}} as the square root of the division \sqrt{\frac{75}{3}} and perform the division.

$$\frac{5}{\frac{\sqrt{37}\times 7\sqrt{37}}{6\times 37}+\sqrt{25}}$$

Calculate the square root of 25 and get 5.

$$\frac{5}{\frac{\sqrt{37}\times 7\sqrt{37}}{6\times 37}+5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{6\times 37}{6\times 37}.

$$\frac{5}{\frac{\sqrt{37}\times 7\sqrt{37}}{6\times 37}+\frac{5\times 6\times 37}{6\times 37}}$$

Since \frac{\sqrt{37}\times 7\sqrt{37}}{6\times 37} and \frac{5\times 6\times 37}{6\times 37} have the same denominator, add them by adding their numerators.

$$\frac{5}{\frac{\sqrt{37}\times 7\sqrt{37}+5\times 6\times 37}{6\times 37}}$$

Do the multiplications in \sqrt{37}\times 7\sqrt{37}+5\times 6\times 37.

$$\frac{5}{\frac{259+1110}{6\times 37}}$$

Do the calculations in 259+1110.

$$\frac{5}{\frac{1369}{6\times 37}}$$

Divide 5 by \frac{1369}{6\times 37} by multiplying 5 by the reciprocal of \frac{1369}{6\times 37}.

$$\frac{5\times 6\times 37}{1369}$$

Multiply 5 and 6 to get 30.

$$\frac{30\times 37}{1369}$$

Multiply 30 and 37 to get 1110.

$$\frac{1110}{1369}$$

Reduce the fraction \frac{1110}{1369} to lowest terms by extracting and canceling out 37.

$$\frac{30}{37}$$

Factor

$$\frac{2 \cdot 3 \cdot 5}{37} = 0.8108108108108109$$