Solve sqrt{left(5-5right)^2+{left(frac{5sqrt{3}}{3}-5sqrt{3}right)}^2}

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

Evaluate

$$\frac{10\sqrt{3}}{3}\approx 5.773502692$$

Solution Steps

Subtract 5 from 5 to get 0.

$$\sqrt{0^{2}+\left(\frac{5\sqrt{3}}{3}-5\sqrt{3}\right)^{2}}$$

Calculate 0 to the power of 2 and get 0.

$$\sqrt{0+\left(\frac{5\sqrt{3}}{3}-5\sqrt{3}\right)^{2}}$$

Combine \frac{5\sqrt{3}}{3} and -5\sqrt{3} to get -\frac{10}{3}\sqrt{3}.

$$\sqrt{0+\left(-\frac{10}{3}\sqrt{3}\right)^{2}}$$

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

$$\sqrt{0+\left(-\frac{10}{3}\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -\frac{10}{3} to the power of 2 and get \frac{100}{9}.

$$\sqrt{0+\frac{100}{9}\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\sqrt{0+\frac{100}{9}\times 3}$$

Multiply \frac{100}{9} and 3 to get \frac{100}{3}.

$$\sqrt{0+\frac{100}{3}}$$

Add 0 and \frac{100}{3} to get \frac{100}{3}.

$$\sqrt{\frac{100}{3}}$$

Rewrite the square root of the division \sqrt{\frac{100}{3}} as the division of square roots \frac{\sqrt{100}}{\sqrt{3}}.

$$\frac{\sqrt{100}}{\sqrt{3}}$$

Calculate the square root of 100 and get 10.

$$\frac{10}{\sqrt{3}}$$

Rationalize the denominator of \frac{10}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

$$\frac{10\sqrt{3}}{3}$$