Using appropriate properties, find `(2)/(3)xx (-5)/(7) +(7)/(3) +(2)/(3) xx (-2)/(7)`.

Answer

To solve the expression \(\frac{2}{3} \times \left(-\frac{5}{7}\right) + \frac{7}{3} + \frac{2}{3} \times \left(-\frac{2}{7}\right)\), we can follow these steps:

### Step 1: Identify and Group Terms
We can group the terms involving \(\frac{2}{3}\) together:
\[ \frac{2}{3} \times \left(-\frac{5}{7}\right) + \frac{2}{3} \times \left(-\frac{2}{7}\right) + \frac{7}{3} \]

### Step 2: Factor Out Common Terms
Using the distributive property, we can factor out \(\frac{2}{3}\) from the first two terms:
\[ \frac{2}{3} \left(-\frac{5}{7} - \frac{2}{7}\right) + \frac{7}{3} \]

### Step 3: Simplify Inside the Parentheses
Now, simplify the expression inside the parentheses:
\[ -\frac{5}{7} - \frac{2}{7} = -\frac{5 + 2}{7} = -\frac{7}{7} = -1 \]
So, we have:
\[ \frac{2}{3} \times (-1) + \frac{7}{3} \]

### Step 4: Multiply and Simplify
Now, multiply:
\[ -\frac{2}{3} + \frac{7}{3} \]

### Step 5: Combine the Terms
Since the denominators are the same, we can combine the numerators:
\[ \frac{-2 + 7}{3} = \frac{5}{3} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{5}{3} \]

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