Solve {left(1/3right)}^2+2(1/3)(1/2)+{left(1/2right)}^2

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

Evaluate

$$\frac{25}{36}\approx 0.694444444$$

Solution Steps

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\frac{1}{9}+2\times \frac{1}{3}\times \frac{1}{2}+\left(\frac{1}{2}\right)^{2}$$

Multiply 2 and \frac{1}{3} to get \frac{2}{3}.

$$\frac{1}{9}+\frac{2}{3}\times \frac{1}{2}+\left(\frac{1}{2}\right)^{2}$$

Multiply \frac{2}{3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{9}+\frac{2\times 1}{3\times 2}+\left(\frac{1}{2}\right)^{2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{1}{9}+\frac{1}{3}+\left(\frac{1}{2}\right)^{2}$$

Least common multiple of 9 and 3 is 9. Convert \frac{1}{9} and \frac{1}{3} to fractions with denominator 9.

$$\frac{1}{9}+\frac{3}{9}+\left(\frac{1}{2}\right)^{2}$$

Since \frac{1}{9} and \frac{3}{9} have the same denominator, add them by adding their numerators.

$$\frac{1+3}{9}+\left(\frac{1}{2}\right)^{2}$$

Add 1 and 3 to get 4.

$$\frac{4}{9}+\left(\frac{1}{2}\right)^{2}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{4}{9}+\frac{1}{4}$$

Least common multiple of 9 and 4 is 36. Convert \frac{4}{9} and \frac{1}{4} to fractions with denominator 36.

$$\frac{16}{36}+\frac{9}{36}$$

Since \frac{16}{36} and \frac{9}{36} have the same denominator, add them by adding their numerators.

$$\frac{16+9}{36}$$

Add 16 and 9 to get 25.

$$\frac{25}{36}$$

Factor

$$\frac{5 ^ {2}}{2 ^ {2} \cdot 3 ^ {2}} = 0.6944444444444444$$