Solve sqrt{11/2}*sqrt{51/4}-sqrt{3}*3

$$\sqrt { 1 \frac { 1 } { 2 } } \times \sqrt { 5 \frac { 1 } { 4 } } - \sqrt { 3 } \cdot 3$$

Evaluate

$$\frac{3\sqrt{14}}{4}-3\sqrt{3}\approx -2.389909383$$

Short Solution Steps

Multiply 1 and 2 to get 2.

$$\sqrt{\frac{2+1}{2}}\sqrt{\frac{5\times 4+1}{4}}-3\sqrt{3}$$

Add 2 and 1 to get 3.

$$\sqrt{\frac{3}{2}}\sqrt{\frac{5\times 4+1}{4}}-3\sqrt{3}$$

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$\frac{\sqrt{3}}{\sqrt{2}}\sqrt{\frac{5\times 4+1}{4}}-3\sqrt{3}$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\sqrt{\frac{5\times 4+1}{4}}-3\sqrt{3}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{\frac{5\times 4+1}{4}}-3\sqrt{3}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{6}}{2}\sqrt{\frac{5\times 4+1}{4}}-3\sqrt{3}$$

Multiply 5 and 4 to get 20.

$$\frac{\sqrt{6}}{2}\sqrt{\frac{20+1}{4}}-3\sqrt{3}$$

Add 20 and 1 to get 21.

$$\frac{\sqrt{6}}{2}\sqrt{\frac{21}{4}}-3\sqrt{3}$$

Rewrite the square root of the division \sqrt{\frac{21}{4}} as the division of square roots \frac{\sqrt{21}}{\sqrt{4}}.

$$\frac{\sqrt{6}}{2}\times \frac{\sqrt{21}}{\sqrt{4}}-3\sqrt{3}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{6}}{2}\times \frac{\sqrt{21}}{2}-3\sqrt{3}$$

Multiply \frac{\sqrt{6}}{2} times \frac{\sqrt{21}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{6}\sqrt{21}}{2\times 2}-3\sqrt{3}$$

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

$$\frac{\sqrt{6}\sqrt{21}}{2\times 2}+\frac{-3\sqrt{3}\times 2\times 2}{2\times 2}$$

Since \frac{\sqrt{6}\sqrt{21}}{2\times 2} and \frac{-3\sqrt{3}\times 2\times 2}{2\times 2} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{6}\sqrt{21}-3\sqrt{3}\times 2\times 2}{2\times 2}$$

Do the multiplications in \sqrt{6}\sqrt{21}-3\sqrt{3}\times 2\times 2.

$$\frac{3\sqrt{14}-12\sqrt{3}}{2\times 2}$$

Expand 2\times 2.

$$\frac{3\sqrt{14}-12\sqrt{3}}{4}$$