Solve sqrt{12}+4sqrt{12/2}-(sqrt{3-9/4}-sqrt{24})

$$\sqrt { 12 } + 4 \sqrt { 1 \frac { 2 } { 2 } } - ( \sqrt { 3 - \frac { 9 } { 4 } } - \sqrt { 24 } )$$

Evaluate

$$\frac{3\sqrt{3}}{2}+2\sqrt{6}+4\sqrt{2}\approx 13.153909946$$

Short Solution Steps

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}.

$$2\sqrt{3}+4\sqrt{\frac{1\times 2+2}{2}}-\left(\sqrt{3-\frac{9}{4}}-\sqrt{24}\right)$$

Multiply 1 and 2 to get 2.

$$2\sqrt{3}+4\sqrt{\frac{2+2}{2}}-\left(\sqrt{3-\frac{9}{4}}-\sqrt{24}\right)$$

Add 2 and 2 to get 4.

$$2\sqrt{3}+4\sqrt{\frac{4}{2}}-\left(\sqrt{3-\frac{9}{4}}-\sqrt{24}\right)$$

Divide 4 by 2 to get 2.

$$2\sqrt{3}+4\sqrt{2}-\left(\sqrt{3-\frac{9}{4}}-\sqrt{24}\right)$$

Convert 3 to fraction \frac{12}{4}.

$$2\sqrt{3}+4\sqrt{2}-\left(\sqrt{\frac{12}{4}-\frac{9}{4}}-\sqrt{24}\right)$$

Since \frac{12}{4} and \frac{9}{4} have the same denominator, subtract them by subtracting their numerators.

$$2\sqrt{3}+4\sqrt{2}-\left(\sqrt{\frac{12-9}{4}}-\sqrt{24}\right)$$

Subtract 9 from 12 to get 3.

$$2\sqrt{3}+4\sqrt{2}-\left(\sqrt{\frac{3}{4}}-\sqrt{24}\right)$$

Rewrite the square root of the division \sqrt{\frac{3}{4}} as the division of square roots \frac{\sqrt{3}}{\sqrt{4}}.

$$2\sqrt{3}+4\sqrt{2}-\left(\frac{\sqrt{3}}{\sqrt{4}}-\sqrt{24}\right)$$

Calculate the square root of 4 and get 2.

$$2\sqrt{3}+4\sqrt{2}-\left(\frac{\sqrt{3}}{2}-\sqrt{24}\right)$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$2\sqrt{3}+4\sqrt{2}-\left(\frac{\sqrt{3}}{2}-2\sqrt{6}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -2\sqrt{6} times \frac{2}{2}.

$$2\sqrt{3}+4\sqrt{2}-\left(\frac{\sqrt{3}}{2}+\frac{2\left(-2\right)\sqrt{6}}{2}\right)$$

Since \frac{\sqrt{3}}{2} and \frac{2\left(-2\right)\sqrt{6}}{2} have the same denominator, add them by adding their numerators.

$$2\sqrt{3}+4\sqrt{2}-\frac{\sqrt{3}+2\left(-2\right)\sqrt{6}}{2}$$

Do the multiplications in \sqrt{3}+2\left(-2\right)\sqrt{6}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{3}+4\sqrt{2} times \frac{2}{2}.

$$\frac{2\left(2\sqrt{3}+4\sqrt{2}\right)}{2}-\frac{\sqrt{3}-4\sqrt{6}}{2}$$

Since \frac{2\left(2\sqrt{3}+4\sqrt{2}\right)}{2} and \frac{\sqrt{3}-4\sqrt{6}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(2\sqrt{3}+4\sqrt{2}\right)-\left(\sqrt{3}-4\sqrt{6}\right)}{2}$$

Do the multiplications in 2\left(2\sqrt{3}+4\sqrt{2}\right)-\left(\sqrt{3}-4\sqrt{6}\right).

$$\frac{4\sqrt{3}+8\sqrt{2}-\sqrt{3}+4\sqrt{6}}{2}$$

Do the calculations in 4\sqrt{3}+8\sqrt{2}-\sqrt{3}+4\sqrt{6}.

$$\frac{3\sqrt{3}+8\sqrt{2}+4\sqrt{6}}{2}$$