Let's break down and simplify each of the given expressions step by step.

First expression:

$- \frac{3}{7} + \frac{5}{13} + \left( - \frac{4}{7} \right)$

1. Combine the fractions with a common denominator:

   • Fractions with denominator 7: $- \frac{3}{7}$ and $- \frac{4}{7}$ $- \frac{3}{7} - \frac{4}{7} = - \frac{7}{7} = -1$ So, the expression becomes: $-1 + \frac{5}{13}$

2. To combine $-1$ and $\frac{5}{13}$, write $-1$ as $- \frac{13}{13}$: $- \frac{13}{13} + \frac{5}{13} = \frac{-13 + 5}{13} = \frac{-8}{13}$

So, the simplified value of the first expression is: $\boxed{- \frac{8}{13}}$

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Second expression:

$\frac{7}{9} + \left( - \frac{1}{3} \right) + \frac{5}{27}$

1. First, convert $- \frac{1}{3}$ to a fraction with denominator 9. Since $3 \times 3 = 9$, we multiply both the numerator and denominator by 3: $- \frac{1}{3} = - \frac{3}{9}$ Now the expression is: $\frac{7}{9} - \frac{3}{9} + \frac{5}{27}$

2. Combine $\frac{7}{9} - \frac{3}{9}$: $\frac{7}{9} - \frac{3}{9} = \frac{4}{9}$ Now the expression becomes: $\frac{4}{9} + \frac{5}{27}$

3. To combine $\frac{4}{9}$ and $\frac{5}{27}$, convert $\frac{4}{9}$ to have a denominator of 27. Multiply both the numerator and denominator by 3: $\frac{4}{9} = \frac{12}{27}$ Now the expression is: $\frac{12}{27} + \frac{5}{27} = \frac{12 + 5}{27} = \frac{17}{27}$

So, the simplified value of the second expression is: $\boxed{\frac{17}{27}}$

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Final Answer:

• First expression: $\boxed{- \frac{8}{13}}$
• Second expression: $\boxed{\frac{17}{27}}$

Let me know if you'd like further explanation or if you have any questions!

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Here are 5 questions related to this:

1. How do you find a common denominator for fractions?
2. What is the process of adding or subtracting fractions with different denominators?
3. How can you simplify a fraction once you've combined terms?
4. Can fractions with different denominators be combined directly without finding a common denominator?
5. How would you handle fractions with large denominators in a similar problem?

Tip: When adding or subtracting fractions, always find a common denominator first, as it allows for straightforward combination of the fractions.