Solve frac{sqrt{75}}{sqrt{12}}+frac{sqrt{108}}{sqrt{141}}-sqrt{48}

$$\frac { \sqrt { 75 } } { \sqrt { 12 } } + \frac { \sqrt { 108 } } { \sqrt { 141 } } - \sqrt { 48 }$$

Evaluate

$$\frac{6\sqrt{47}}{47}+\frac{5}{2}-4\sqrt{3}\approx -3.553013281$$

Solution Steps

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}.

$$\frac{5\sqrt{3}}{\sqrt{12}}+\frac{\sqrt{108}}{\sqrt{141}}-\sqrt{48}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{5\sqrt{3}}{2\sqrt{3}}+\frac{\sqrt{108}}{\sqrt{141}}-\sqrt{48}$$

Cancel out \sqrt{3} in both numerator and denominator.

$$\frac{5}{2}+\frac{\sqrt{108}}{\sqrt{141}}-\sqrt{48}$$

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}.

$$\frac{5}{2}+\frac{6\sqrt{3}}{\sqrt{141}}-\sqrt{48}$$

Rationalize the denominator of \frac{6\sqrt{3}}{\sqrt{141}} by multiplying numerator and denominator by \sqrt{141}.

$$\frac{5}{2}+\frac{6\sqrt{3}\sqrt{141}}{\left(\sqrt{141}\right)^{2}}-\sqrt{48}$$

The square of \sqrt{141} is 141.

$$\frac{5}{2}+\frac{6\sqrt{3}\sqrt{141}}{141}-\sqrt{48}$$

Factor 141=3\times 47. Rewrite the square root of the product \sqrt{3\times 47} as the product of square roots \sqrt{3}\sqrt{47}.

$$\frac{5}{2}+\frac{6\sqrt{3}\sqrt{3}\sqrt{47}}{141}-\sqrt{48}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{5}{2}+\frac{6\times 3\sqrt{47}}{141}-\sqrt{48}$$

Multiply 6 and 3 to get 18.

$$\frac{5}{2}+\frac{18\sqrt{47}}{141}-\sqrt{48}$$

Divide 18\sqrt{47} by 141 to get \frac{6}{47}\sqrt{47}.

$$\frac{5}{2}+\frac{6}{47}\sqrt{47}-\sqrt{48}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{5}{2}+\frac{6}{47}\sqrt{47}-4\sqrt{3}$$