Solve sqrt{30*6400*0.3704/270^2+100^2}

$$\sqrt{ \frac{ 30 \cdot 6400 \cdot 0.3704 }{ { 270 }^{ 2 } + { 100 }^{ 2 } } }$$

Evaluate

$$\frac{8\sqrt{5757405}}{20725}\approx 0.926208662$$

Solution Steps

Multiply 30 and 6400 to get 192000.

$$\sqrt{\frac{192000\times 0.3704}{270^{2}+100^{2}}}$$

Multiply 192000 and 0.3704 to get 71116.8.

$$\sqrt{\frac{71116.8}{270^{2}+100^{2}}}$$

Calculate 270 to the power of 2 and get 72900.

$$\sqrt{\frac{71116.8}{72900+100^{2}}}$$

Calculate 100 to the power of 2 and get 10000.

$$\sqrt{\frac{71116.8}{72900+10000}}$$

Add 72900 and 10000 to get 82900.

$$\sqrt{\frac{71116.8}{82900}}$$

Expand \frac{71116.8}{82900} by multiplying both numerator and the denominator by 10.

$$\sqrt{\frac{711168}{829000}}$$

Reduce the fraction \frac{711168}{829000} to lowest terms by extracting and canceling out 8.

$$\sqrt{\frac{88896}{103625}}$$

Rewrite the square root of the division \sqrt{\frac{88896}{103625}} as the division of square roots \frac{\sqrt{88896}}{\sqrt{103625}}.

$$\frac{\sqrt{88896}}{\sqrt{103625}}$$

Factor 88896=8^{2}\times 1389. Rewrite the square root of the product \sqrt{8^{2}\times 1389} as the product of square roots \sqrt{8^{2}}\sqrt{1389}. Take the square root of 8^{2}.

$$\frac{8\sqrt{1389}}{\sqrt{103625}}$$

Factor 103625=5^{2}\times 4145. Rewrite the square root of the product \sqrt{5^{2}\times 4145} as the product of square roots \sqrt{5^{2}}\sqrt{4145}. Take the square root of 5^{2}.

$$\frac{8\sqrt{1389}}{5\sqrt{4145}}$$

Rationalize the denominator of \frac{8\sqrt{1389}}{5\sqrt{4145}} by multiplying numerator and denominator by \sqrt{4145}.

$$\frac{8\sqrt{1389}\sqrt{4145}}{5\left(\sqrt{4145}\right)^{2}}$$

The square of \sqrt{4145} is 4145.

$$\frac{8\sqrt{1389}\sqrt{4145}}{5\times 4145}$$

To multiply \sqrt{1389} and \sqrt{4145}, multiply the numbers under the square root.

$$\frac{8\sqrt{5757405}}{5\times 4145}$$

Multiply 5 and 4145 to get 20725.

$$\frac{8\sqrt{5757405}}{20725}$$