Solve sqrt{300^2/23+100^2/23}

$$\sqrt { \frac { 300 ^ { 2 } } { 23 } + \frac { 100 ^ { 2 } } { 23 } }$$

Evaluate

$$\frac{100\sqrt{230}}{23}\approx 65.93804734$$

Solution Steps

Calculate 300 to the power of 2 and get 90000.

$$\sqrt{\frac{90000}{23}+\frac{100^{2}}{23}}$$

Calculate 100 to the power of 2 and get 10000.

$$\sqrt{\frac{90000}{23}+\frac{10000}{23}}$$

Since \frac{90000}{23} and \frac{10000}{23} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{90000+10000}{23}}$$

Add 90000 and 10000 to get 100000.

$$\sqrt{\frac{100000}{23}}$$

Rewrite the square root of the division \sqrt{\frac{100000}{23}} as the division of square roots \frac{\sqrt{100000}}{\sqrt{23}}.

$$\frac{\sqrt{100000}}{\sqrt{23}}$$

Factor 100000=100^{2}\times 10. Rewrite the square root of the product \sqrt{100^{2}\times 10} as the product of square roots \sqrt{100^{2}}\sqrt{10}. Take the square root of 100^{2}.

$$\frac{100\sqrt{10}}{\sqrt{23}}$$

Rationalize the denominator of \frac{100\sqrt{10}}{\sqrt{23}} by multiplying numerator and denominator by \sqrt{23}.

$$\frac{100\sqrt{10}\sqrt{23}}{\left(\sqrt{23}\right)^{2}}$$

The square of \sqrt{23} is 23.

$$\frac{100\sqrt{10}\sqrt{23}}{23}$$

To multiply \sqrt{10} and \sqrt{23}, multiply the numbers under the square root.

$$\frac{100\sqrt{230}}{23}$$