Solve sqrt{75+(45^2+40)/65*10^4}

$$\sqrt { \frac { 75 + ( 45 ^ { 2 } + 40 ) } { 65 \times 10 ^ { 4 } } }$$

Evaluate

$$\frac{\sqrt{1391}}{650}\approx 0.057378634$$

Solution Steps

Calculate 45 to the power of 2 and get 2025.

$$\sqrt{\frac{75+2025+40}{65\times 10^{4}}}$$

Add 75 and 2025 to get 2100.

$$\sqrt{\frac{2100+40}{65\times 10^{4}}}$$

Add 2100 and 40 to get 2140.

$$\sqrt{\frac{2140}{65\times 10^{4}}}$$

Calculate 10 to the power of 4 and get 10000.

$$\sqrt{\frac{2140}{65\times 10000}}$$

Multiply 65 and 10000 to get 650000.

$$\sqrt{\frac{2140}{650000}}$$

Reduce the fraction \frac{2140}{650000} to lowest terms by extracting and canceling out 20.

$$\sqrt{\frac{107}{32500}}$$

Rewrite the square root of the division \sqrt{\frac{107}{32500}} as the division of square roots \frac{\sqrt{107}}{\sqrt{32500}}.

$$\frac{\sqrt{107}}{\sqrt{32500}}$$

Factor 32500=50^{2}\times 13. Rewrite the square root of the product \sqrt{50^{2}\times 13} as the product of square roots \sqrt{50^{2}}\sqrt{13}. Take the square root of 50^{2}.

$$\frac{\sqrt{107}}{50\sqrt{13}}$$

Rationalize the denominator of \frac{\sqrt{107}}{50\sqrt{13}} by multiplying numerator and denominator by \sqrt{13}.

$$\frac{\sqrt{107}\sqrt{13}}{50\left(\sqrt{13}\right)^{2}}$$

The square of \sqrt{13} is 13.

$$\frac{\sqrt{107}\sqrt{13}}{50\times 13}$$

To multiply \sqrt{107} and \sqrt{13}, multiply the numbers under the square root.

$$\frac{\sqrt{1391}}{50\times 13}$$

Multiply 50 and 13 to get 650.

$$\frac{\sqrt{1391}}{650}$$