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

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

Evaluate

$$\frac{5\sqrt{2}}{3}\approx 2.357022604$$

Solution Steps

Calculate \frac{4}{3} to the power of 2 and get \frac{16}{9}.

$$\sqrt{\frac{16}{9}+2\times \frac{4}{3}+1+\left(\frac{1}{3}\right)^{2}}$$

Express 2\times \frac{4}{3} as a single fraction.

$$\sqrt{\frac{16}{9}+\frac{2\times 4}{3}+1+\left(\frac{1}{3}\right)^{2}}$$

Multiply 2 and 4 to get 8.

$$\sqrt{\frac{16}{9}+\frac{8}{3}+1+\left(\frac{1}{3}\right)^{2}}$$

Least common multiple of 9 and 3 is 9. Convert \frac{16}{9} and \frac{8}{3} to fractions with denominator 9.

$$\sqrt{\frac{16}{9}+\frac{24}{9}+1+\left(\frac{1}{3}\right)^{2}}$$

Since \frac{16}{9} and \frac{24}{9} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{16+24}{9}+1+\left(\frac{1}{3}\right)^{2}}$$

Add 16 and 24 to get 40.

$$\sqrt{\frac{40}{9}+1+\left(\frac{1}{3}\right)^{2}}$$

Convert 1 to fraction \frac{9}{9}.

$$\sqrt{\frac{40}{9}+\frac{9}{9}+\left(\frac{1}{3}\right)^{2}}$$

Since \frac{40}{9} and \frac{9}{9} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{40+9}{9}+\left(\frac{1}{3}\right)^{2}}$$

Add 40 and 9 to get 49.

$$\sqrt{\frac{49}{9}+\left(\frac{1}{3}\right)^{2}}$$

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

$$\sqrt{\frac{49}{9}+\frac{1}{9}}$$

Since \frac{49}{9} and \frac{1}{9} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{49+1}{9}}$$

Add 49 and 1 to get 50.

$$\sqrt{\frac{50}{9}}$$

Rewrite the square root of the division \sqrt{\frac{50}{9}} as the division of square roots \frac{\sqrt{50}}{\sqrt{9}}.

$$\frac{\sqrt{50}}{\sqrt{9}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\frac{5\sqrt{2}}{\sqrt{9}}$$

Calculate the square root of 9 and get 3.

$$\frac{5\sqrt{2}}{3}$$