Solve sqrt{13.5div4/3div3.14}

$$\sqrt{ 13.5 \div \frac{ 4 }{ 3 } \div 3.14 }$$

Evaluate

$$\frac{45\sqrt{157}}{314}\approx 1.79569549$$

Solution Steps

Express \frac{\frac{13.5}{\frac{4}{3}}}{3.14} as a single fraction.

$$\sqrt{\frac{13.5}{\frac{4}{3}\times 3.14}}$$

Convert decimal number 3.14 to fraction \frac{314}{100}. Reduce the fraction \frac{314}{100} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{13.5}{\frac{4}{3}\times \frac{157}{50}}}$$

Multiply \frac{4}{3} times \frac{157}{50} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{13.5}{\frac{4\times 157}{3\times 50}}}$$

Do the multiplications in the fraction \frac{4\times 157}{3\times 50}.

$$\sqrt{\frac{13.5}{\frac{628}{150}}}$$

Reduce the fraction \frac{628}{150} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{13.5}{\frac{314}{75}}}$$

Divide 13.5 by \frac{314}{75} by multiplying 13.5 by the reciprocal of \frac{314}{75}.

$$\sqrt{13.5\times \frac{75}{314}}$$

Convert decimal number 13.5 to fraction \frac{135}{10}. Reduce the fraction \frac{135}{10} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{27}{2}\times \frac{75}{314}}$$

Multiply \frac{27}{2} times \frac{75}{314} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{27\times 75}{2\times 314}}$$

Do the multiplications in the fraction \frac{27\times 75}{2\times 314}.

$$\sqrt{\frac{2025}{628}}$$

Rewrite the square root of the division \sqrt{\frac{2025}{628}} as the division of square roots \frac{\sqrt{2025}}{\sqrt{628}}.

$$\frac{\sqrt{2025}}{\sqrt{628}}$$

Calculate the square root of 2025 and get 45.

$$\frac{45}{\sqrt{628}}$$

Factor 628=2^{2}\times 157. Rewrite the square root of the product \sqrt{2^{2}\times 157} as the product of square roots \sqrt{2^{2}}\sqrt{157}. Take the square root of 2^{2}.

$$\frac{45}{2\sqrt{157}}$$

Rationalize the denominator of \frac{45}{2\sqrt{157}} by multiplying numerator and denominator by \sqrt{157}.

$$\frac{45\sqrt{157}}{2\left(\sqrt{157}\right)^{2}}$$

The square of \sqrt{157} is 157.

$$\frac{45\sqrt{157}}{2\times 157}$$

Multiply 2 and 157 to get 314.

$$\frac{45\sqrt{157}}{314}$$