Solve sqrt{(frac{10sqrt{3}}{3})^2+25^2}

$$\sqrt { ( \frac { 10 \sqrt { 3 } } { 3 } ) ^ { 2 } + 25 ^ { 2 } }$$

Evaluate

$$\frac{5\sqrt{237}}{3}\approx 25.658007197$$

Solution Steps

To raise \frac{10\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(10\sqrt{3}\right)^{2}}{3^{2}}+25^{2}}$$

Calculate 25 to the power of 2 and get 625.

$$\sqrt{\frac{\left(10\sqrt{3}\right)^{2}}{3^{2}}+625}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 625 times \frac{3^{2}}{3^{2}}.

$$\sqrt{\frac{\left(10\sqrt{3}\right)^{2}}{3^{2}}+\frac{625\times 3^{2}}{3^{2}}}$$

Since \frac{\left(10\sqrt{3}\right)^{2}}{3^{2}} and \frac{625\times 3^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\left(10\sqrt{3}\right)^{2}+625\times 3^{2}}{3^{2}}}$$

Expand \left(10\sqrt{3}\right)^{2}.

$$\sqrt{\frac{10^{2}\left(\sqrt{3}\right)^{2}+625\times 3^{2}}{3^{2}}}$$

Calculate 10 to the power of 2 and get 100.

$$\sqrt{\frac{100\left(\sqrt{3}\right)^{2}+625\times 3^{2}}{3^{2}}}$$

The square of \sqrt{3} is 3.

$$\sqrt{\frac{100\times 3+625\times 3^{2}}{3^{2}}}$$

Multiply 100 and 3 to get 300.

$$\sqrt{\frac{300+625\times 3^{2}}{3^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{\frac{300+625\times 9}{3^{2}}}$$

Multiply 625 and 9 to get 5625.

$$\sqrt{\frac{300+5625}{3^{2}}}$$

Add 300 and 5625 to get 5925.

$$\sqrt{\frac{5925}{3^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{\frac{5925}{9}}$$

Reduce the fraction \frac{5925}{9} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{1975}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1975}{3}} as the division of square roots \frac{\sqrt{1975}}{\sqrt{3}}.

$$\frac{\sqrt{1975}}{\sqrt{3}}$$

Factor 1975=5^{2}\times 79. Rewrite the square root of the product \sqrt{5^{2}\times 79} as the product of square roots \sqrt{5^{2}}\sqrt{79}. Take the square root of 5^{2}.

$$\frac{5\sqrt{79}}{\sqrt{3}}$$

Rationalize the denominator of \frac{5\sqrt{79}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{5\sqrt{79}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5\sqrt{79}\sqrt{3}}{3}$$

To multiply \sqrt{79} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{5\sqrt{237}}{3}$$