Solve 2sqrt{27}-frac{7sqrt{5}}{sqrt{15}}+sqrt{108}-frac{sqrt{75}}{3}

$$2 \sqrt{ 27 } - \frac{ 7 \sqrt{ 5 } }{ \sqrt{ 15 } } + \sqrt{ 108 } - \frac{ \sqrt{ 75 } }{ 3 }$$

Evaluate

$$8\sqrt{3}\approx 13.856406461$$

Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$2\times 3\sqrt{3}-\frac{7\sqrt{5}}{\sqrt{15}}+\sqrt{108}-\frac{\sqrt{75}}{3}$$

Multiply 2 and 3 to get 6.

$$6\sqrt{3}-\frac{7\sqrt{5}}{\sqrt{15}}+\sqrt{108}-\frac{\sqrt{75}}{3}$$

Rationalize the denominator of \frac{7\sqrt{5}}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$6\sqrt{3}-\frac{7\sqrt{5}\sqrt{15}}{\left(\sqrt{15}\right)^{2}}+\sqrt{108}-\frac{\sqrt{75}}{3}$$

The square of \sqrt{15} is 15.

$$6\sqrt{3}-\frac{7\sqrt{5}\sqrt{15}}{15}+\sqrt{108}-\frac{\sqrt{75}}{3}$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

$$6\sqrt{3}-\frac{7\sqrt{5}\sqrt{5}\sqrt{3}}{15}+\sqrt{108}-\frac{\sqrt{75}}{3}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$6\sqrt{3}-\frac{7\times 5\sqrt{3}}{15}+\sqrt{108}-\frac{\sqrt{75}}{3}$$

Multiply 7 and 5 to get 35.

$$6\sqrt{3}-\frac{35\sqrt{3}}{15}+\sqrt{108}-\frac{\sqrt{75}}{3}$$

Divide 35\sqrt{3} by 15 to get \frac{7}{3}\sqrt{3}.

$$6\sqrt{3}-\frac{7}{3}\sqrt{3}+\sqrt{108}-\frac{\sqrt{75}}{3}$$

Factor 108=6^{2}\times 3. Rewrite the square root of the product \sqrt{6^{2}\times 3} as the product of square roots \sqrt{6^{2}}\sqrt{3}. Take the square root of 6^{2}.

$$6\sqrt{3}-\frac{7}{3}\sqrt{3}+6\sqrt{3}-\frac{\sqrt{75}}{3}$$

Combine 6\sqrt{3} and 6\sqrt{3} to get 12\sqrt{3}.

$$12\sqrt{3}-\frac{7}{3}\sqrt{3}-\frac{\sqrt{75}}{3}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$12\sqrt{3}-\frac{7}{3}\sqrt{3}-\frac{5\sqrt{3}}{3}$$

Combine 12\sqrt{3} and -\frac{5\sqrt{3}}{3} to get \frac{31}{3}\sqrt{3}.

$$\frac{31}{3}\sqrt{3}-\frac{7}{3}\sqrt{3}$$

Combine \frac{31}{3}\sqrt{3} and -\frac{7}{3}\sqrt{3} to get 8\sqrt{3}.

$$8\sqrt{3}$$