Solve (1+frac{sqrt{2}}{2})^2+(frac{sqrt{2}}{2})^2

$$( 1 + \frac { \sqrt { 2 } } { 2 } ) ^ { 2 } + ( \frac { \sqrt { 2 } } { 2 } ) ^ { 2 }$$

Evaluate

$$\sqrt{2}+2\approx 3.414213562$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

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

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

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

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

Add 4 and 2 to get 6.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

$$\frac{6+4\sqrt{2}}{4}+\frac{2}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

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

Since \frac{6+4\sqrt{2}}{4} and \frac{2}{4} have the same denominator, add them by adding their numerators.

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

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

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

Divide each term of 8+4\sqrt{2} by 4 to get 2+\sqrt{2}.

$$2+\sqrt{2}$$

Expand

$$\sqrt{2} + 2 = 3.414213562$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

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

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

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

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

Add 4 and 2 to get 6.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

$$\frac{6+4\sqrt{2}}{4}+\frac{2}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

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

Since \frac{6+4\sqrt{2}}{4} and \frac{2}{4} have the same denominator, add them by adding their numerators.

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

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

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

Divide each term of 8+4\sqrt{2} by 4 to get 2+\sqrt{2}.

$$2+\sqrt{2}$$