Solve 10sqrt{27/4}-5sqrt{243/16}-2sqrt{3}

$$10 \sqrt { \frac { 27 } { 4 } } - 5 \sqrt { \frac { 243 } { 16 } } - 2 \sqrt { 3 }$$

Evaluate

$$\frac{7\sqrt{3}}{4}\approx 3.031088913$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{27}{4}} as the division of square roots \frac{\sqrt{27}}{\sqrt{4}}.

$$10\times \frac{\sqrt{27}}{\sqrt{4}}-5\sqrt{\frac{243}{16}}-2\sqrt{3}$$

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}.

$$10\times \frac{3\sqrt{3}}{\sqrt{4}}-5\sqrt{\frac{243}{16}}-2\sqrt{3}$$

Calculate the square root of 4 and get 2.

$$10\times \frac{3\sqrt{3}}{2}-5\sqrt{\frac{243}{16}}-2\sqrt{3}$$

Cancel out 2, the greatest common factor in 10 and 2.

$$5\times 3\sqrt{3}-5\sqrt{\frac{243}{16}}-2\sqrt{3}$$

Rewrite the square root of the division \sqrt{\frac{243}{16}} as the division of square roots \frac{\sqrt{243}}{\sqrt{16}}.

$$5\times 3\sqrt{3}-5\times \frac{\sqrt{243}}{\sqrt{16}}-2\sqrt{3}$$

Factor 243=9^{2}\times 3. Rewrite the square root of the product \sqrt{9^{2}\times 3} as the product of square roots \sqrt{9^{2}}\sqrt{3}. Take the square root of 9^{2}.

$$5\times 3\sqrt{3}-5\times \frac{9\sqrt{3}}{\sqrt{16}}-2\sqrt{3}$$

Calculate the square root of 16 and get 4.

$$5\times 3\sqrt{3}-5\times \frac{9\sqrt{3}}{4}-2\sqrt{3}$$

Express -5\times \frac{9\sqrt{3}}{4} as a single fraction.

$$5\times 3\sqrt{3}+\frac{-5\times 9\sqrt{3}}{4}-2\sqrt{3}$$

Multiply 5 and 3 to get 15.

$$15\sqrt{3}+\frac{-5\times 9\sqrt{3}}{4}-2\sqrt{3}$$

Multiply -5 and 9 to get -45.

$$15\sqrt{3}+\frac{-45\sqrt{3}}{4}-2\sqrt{3}$$

Combine 15\sqrt{3} and \frac{-45\sqrt{3}}{4} to get \frac{15}{4}\sqrt{3}.

$$\frac{15}{4}\sqrt{3}-2\sqrt{3}$$

Combine \frac{15}{4}\sqrt{3} and -2\sqrt{3} to get \frac{7}{4}\sqrt{3}.

$$\frac{7}{4}\sqrt{3}$$