Solve (frac{sqrt{6}+3sqrt{2}-2sqrt{3}}{2})^2

$$( \frac { \sqrt { 6 } + 3 \sqrt { 2 } - 2 \sqrt { 3 } } { 2 } ) ^ { 2 }$$

Evaluate

$$3\sqrt{3}+9-3\sqrt{2}-3\sqrt{6}\approx 2.605042507$$

Solution Steps

To raise \frac{\sqrt{6}+3\sqrt{2}-2\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Square \sqrt{6}+3\sqrt{2}-2\sqrt{3}.

$$\frac{6\sqrt{2}\sqrt{6}-12\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

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

$$\frac{6\sqrt{2}\sqrt{2}\sqrt{3}-12\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{6\times 2\sqrt{3}-12\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply 6 and 2 to get 12.

$$\frac{12\sqrt{3}-12\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

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

$$\frac{12\sqrt{3}-12\sqrt{6}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

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

$$\frac{12\sqrt{3}-12\sqrt{6}-4\sqrt{3}\sqrt{3}\sqrt{2}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

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

$$\frac{12\sqrt{3}-12\sqrt{6}-4\times 3\sqrt{2}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply -4 and 3 to get -12.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+4\times 3+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+12+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+12+9\times 2+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+12+18+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Add 12 and 18 to get 30.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+30+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+30+6}{2^{2}}$$

Add 30 and 6 to get 36.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+36}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+36}{4}$$

Expand

$$3 \sqrt{3} + 9 - 3 \sqrt{2} - 3 \sqrt{6} = 2.605042507$$

Solution Steps

To raise \frac{\sqrt{6}+3\sqrt{2}-2\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Square \sqrt{6}+3\sqrt{2}-2\sqrt{3}.

$$\frac{6\sqrt{2}\sqrt{6}-12\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

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

$$\frac{6\sqrt{2}\sqrt{2}\sqrt{3}-12\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{6\times 2\sqrt{3}-12\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply 6 and 2 to get 12.

$$\frac{12\sqrt{3}-12\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

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

$$\frac{12\sqrt{3}-12\sqrt{6}-4\sqrt{3}\sqrt{6}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

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

$$\frac{12\sqrt{3}-12\sqrt{6}-4\sqrt{3}\sqrt{3}\sqrt{2}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

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

$$\frac{12\sqrt{3}-12\sqrt{6}-4\times 3\sqrt{2}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply -4 and 3 to get -12.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+4\left(\sqrt{3}\right)^{2}+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+4\times 3+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+12+9\left(\sqrt{2}\right)^{2}+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+12+9\times 2+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+12+18+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

Add 12 and 18 to get 30.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+30+\left(\sqrt{6}\right)^{2}}{2^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+30+6}{2^{2}}$$

Add 30 and 6 to get 36.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+36}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{12\sqrt{3}-12\sqrt{6}-12\sqrt{2}+36}{4}$$