Solve sqrt{12}*1/4sqrt{3}divsqrt{2}+(1-sqrt{2})^2

$$\sqrt { 12 } \times \frac { 1 } { 4 } \sqrt { 3 } \div \sqrt { 2 } + ( 1 - \sqrt { 2 } ) ^ { 2 }$$

Evaluate

$$-\frac{5\sqrt{2}}{4}+3\approx 1.232233047$$

Solution Steps

Factor 12=3\times 4. Rewrite the square root of the product \sqrt{3\times 4} as the product of square roots \sqrt{3}\sqrt{4}.

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

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

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

Multiply 3 and \frac{1}{4} to get \frac{3}{4}.

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

Calculate the square root of 4 and get 2.

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

Multiply \frac{3}{4} and 2 to get \frac{3}{2}.

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

Express \frac{\frac{3}{2}}{\sqrt{2}} as a single fraction.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{2}\right)^{2}.

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

The square of \sqrt{2} is 2.

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

Add 1 and 2 to get 3.

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

Combine \frac{3\sqrt{2}}{4} and -2\sqrt{2} to get -\frac{5}{4}\sqrt{2}.

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

Factor

$$\frac{12 - 5 \sqrt{2}}{4} = 1.2322330470336311$$