Solve |-2/3*3+(-3)+5/3|/sqrt{(-frac{2{3})^2+1^2}}

$$\frac { | - \frac { 2 } { 3 } \cdot 3 + ( - 3 ) + \frac { 5 } { 3 } | } { \sqrt { ( - \frac { 2 } { 3 } ) ^ { 2 } + 1 ^ { 2 } } }$$

Evaluate

$$\frac{10\sqrt{13}}{13}\approx 2.773500981$$

Solution Steps

Cancel out 3 and 3.

$$\frac{|-2-3+\frac{5}{3}|}{\sqrt{\left(-\frac{2}{3}\right)^{2}+1^{2}}}$$

Subtract 3 from -2 to get -5.

$$\frac{|-5+\frac{5}{3}|}{\sqrt{\left(-\frac{2}{3}\right)^{2}+1^{2}}}$$

Convert -5 to fraction -\frac{15}{3}.

$$\frac{|-\frac{15}{3}+\frac{5}{3}|}{\sqrt{\left(-\frac{2}{3}\right)^{2}+1^{2}}}$$

Since -\frac{15}{3} and \frac{5}{3} have the same denominator, add them by adding their numerators.

$$\frac{|\frac{-15+5}{3}|}{\sqrt{\left(-\frac{2}{3}\right)^{2}+1^{2}}}$$

Add -15 and 5 to get -10.

$$\frac{|-\frac{10}{3}|}{\sqrt{\left(-\frac{2}{3}\right)^{2}+1^{2}}}$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -\frac{10}{3} is \frac{10}{3}.

$$\frac{\frac{10}{3}}{\sqrt{\left(-\frac{2}{3}\right)^{2}+1^{2}}}$$

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

$$\frac{\frac{10}{3}}{\sqrt{\frac{4}{9}+1^{2}}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\frac{10}{3}}{\sqrt{\frac{4}{9}+1}}$$

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

$$\frac{\frac{10}{3}}{\sqrt{\frac{4}{9}+\frac{9}{9}}}$$

Since \frac{4}{9} and \frac{9}{9} have the same denominator, add them by adding their numerators.

$$\frac{\frac{10}{3}}{\sqrt{\frac{4+9}{9}}}$$

Add 4 and 9 to get 13.

$$\frac{\frac{10}{3}}{\sqrt{\frac{13}{9}}}$$

Rewrite the square root of the division \sqrt{\frac{13}{9}} as the division of square roots \frac{\sqrt{13}}{\sqrt{9}}.

$$\frac{\frac{10}{3}}{\frac{\sqrt{13}}{\sqrt{9}}}$$

Calculate the square root of 9 and get 3.

$$\frac{\frac{10}{3}}{\frac{\sqrt{13}}{3}}$$

Divide \frac{10}{3} by \frac{\sqrt{13}}{3} by multiplying \frac{10}{3} by the reciprocal of \frac{\sqrt{13}}{3}.

$$\frac{10\times 3}{3\sqrt{13}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{10}{\sqrt{13}}$$

Rationalize the denominator of \frac{10}{\sqrt{13}} by multiplying numerator and denominator by \sqrt{13}.

$$\frac{10\sqrt{13}}{\left(\sqrt{13}\right)^{2}}$$

The square of \sqrt{13} is 13.

$$\frac{10\sqrt{13}}{13}$$