Solve 1/84sqrt{500*400/3.14*0.029*0.029*7.85}

$$\frac { 1 } { 84 } \sqrt { \frac { 500 \times 400 } { 3.14 \times 0.029 \times 0.029 \times 7.85 } }$$

Evaluate

$$\frac{2500000\sqrt{2}}{95613}\approx 36.977543911$$

Solution Steps

Multiply 500 and 400 to get 200000.

$$\frac{1}{84}\sqrt{\frac{200000}{3.14\times 0.029\times 0.029\times 7.85}}$$

Multiply 3.14 and 0.029 to get 0.09106.

$$\frac{1}{84}\sqrt{\frac{200000}{0.09106\times 0.029\times 7.85}}$$

Multiply 0.09106 and 0.029 to get 0.00264074.

$$\frac{1}{84}\sqrt{\frac{200000}{0.00264074\times 7.85}}$$

Multiply 0.00264074 and 7.85 to get 0.020729809.

$$\frac{1}{84}\sqrt{\frac{200000}{0.020729809}}$$

Expand \frac{200000}{0.020729809} by multiplying both numerator and the denominator by 1000000000.

$$\frac{1}{84}\sqrt{\frac{200000000000000}{20729809}}$$

Rewrite the square root of the division \sqrt{\frac{200000000000000}{20729809}} as the division of square roots \frac{\sqrt{200000000000000}}{\sqrt{20729809}}.

$$\frac{1}{84}\times \frac{\sqrt{200000000000000}}{\sqrt{20729809}}$$

Factor 200000000000000=10000000^{2}\times 2. Rewrite the square root of the product \sqrt{10000000^{2}\times 2} as the product of square roots \sqrt{10000000^{2}}\sqrt{2}. Take the square root of 10000000^{2}.

$$\frac{1}{84}\times \frac{10000000\sqrt{2}}{\sqrt{20729809}}$$

Calculate the square root of 20729809 and get 4553.

$$\frac{1}{84}\times \frac{10000000\sqrt{2}}{4553}$$

Multiply \frac{1}{84} times \frac{10000000\sqrt{2}}{4553} by multiplying numerator times numerator and denominator times denominator.

$$\frac{10000000\sqrt{2}}{84\times 4553}$$

Cancel out 4 in both numerator and denominator.

$$\frac{2500000\sqrt{2}}{21\times 4553}$$

Multiply 21 and 4553 to get 95613.

$$\frac{2500000\sqrt{2}}{95613}$$