Let's break down each of these expressions and simplify them one by one.

1. Expression for a:

$3a, \quad \frac{7}{21} + \frac{9}{-36}$ Simplify the fractions: $\frac{7}{21} = \frac{1}{3}, \quad \frac{9}{-36} = \frac{-1}{4}$ So, the expression becomes: $\frac{1}{3} + \frac{-1}{4}$ To add these, find a common denominator (12): $\frac{1}{3} = \frac{4}{12}, \quad \frac{-1}{4} = \frac{-3}{12}$ Now, add the fractions: $\frac{4}{12} + \frac{-3}{12} = \frac{1}{12}$

2. Expression for b:

$b, \quad \frac{-3}{29} + \frac{16}{58}$ Note that $\frac{16}{58}$ can be simplified: $\frac{16}{58} = \frac{8}{29}$ So, the expression becomes: $\frac{-3}{29} + \frac{8}{29}$ Now, add the fractions: $\frac{-3 + 8}{29} = \frac{5}{29}$

3. Expression for -6/13 + [1 + -7/13]:

$\frac{-6}{13} + \left(1 + \frac{-7}{13}\right)$ Simplify inside the parentheses: $1 + \frac{-7}{13} = \frac{13}{13} - \frac{7}{13} = \frac{6}{13}$ Now, add: $\frac{-6}{13} + \frac{6}{13} = 0$

4. Expression for d:

$\frac{5}{17} + \frac{-9}{15} + \frac{-2}{17} + \frac{2}{-5}$ Simplify step-by-step:

• $\frac{-9}{15} = \frac{-3}{5}$
• $\frac{5}{17} - \frac{2}{17} = \frac{3}{17}$

Now, the expression becomes: $\frac{3}{17} + \frac{-3}{5} + \frac{2}{-5}$ Combine the fractions involving 5: $\frac{-3}{5} + \frac{-2}{5} = \frac{-5}{5} = -1$ Now, the expression is: $\frac{3}{17} - 1$ Write 1 as $\frac{17}{17}$, and subtract: $\frac{3}{17} - \frac{17}{17} = \frac{-14}{17}$

5. Expression for 4a:

$\frac{4}{13} + \frac{-12}{39}$ Simplify $\frac{-12}{39}$: $\frac{-12}{39} = \frac{-4}{13}$ Now, add the fractions: $\frac{4}{13} + \frac{-4}{13} = 0$

6. Expression for b:

$\frac{-8}{18} + \frac{-15}{27}$ Simplify: $\frac{-8}{18} = \frac{-4}{9}, \quad \frac{-15}{27} = \frac{-5}{9}$ Now, add the fractions: $\frac{-4}{9} + \frac{-5}{9} = \frac{-9}{9} = -1$

7. Expression for c:

$\frac{-7}{31} + \left(99 + \frac{7}{31}\right)$ Simplify inside the parentheses: $99 + \frac{7}{31} = \frac{3069}{31} + \frac{7}{31} = \frac{3076}{31}$ Now, subtract: $\frac{-7}{31} + \frac{3076}{31} = \frac{3076 - 7}{31} = \frac{3069}{31} = 99$

8. Expression for d:

$\frac{-1}{2} + \frac{7}{21} + \frac{-2}{6} + \frac{-5}{30}$ Simplify the fractions: $\frac{7}{21} = \frac{1}{3}, \quad \frac{-2}{6} = \frac{-1}{3}, \quad \frac{-5}{30} = \frac{-1}{6}$ Now, add them step-by-step: $\frac{-1}{2} + \frac{1}{3} = \frac{-3}{6} + \frac{2}{6} = \frac{-1}{6}$ $\frac{-1}{6} + \frac{-1}{3} = \frac{-1}{6} + \frac{-2}{6} = \frac{-3}{6} = \frac{-1}{2}$ $\frac{-1}{2} + \frac{-1}{6} = \frac{-3}{6} + \frac{-1}{6} = \frac{-4}{6} = \frac{-2}{3}$

Final Results:

• a: $\frac{1}{12}$
• b: $\frac{5}{29}$
• -6/13 + [1 + -7/13]: $0$
• d: $\frac{-14}{17}$
• 4a: $0$
• b: $-1$
• c: $99$
• d: $\frac{-2}{3}$

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Would you like further explanations for any of these steps, or have any questions about the solutions?

Related Questions:

1. How to simplify expressions with common denominators?
2. What is the process for handling mixed fractions in calculations?
3. How do you deal with negative fractions in addition or subtraction?
4. What’s the best approach for simplifying complex rational expressions?
5. How to find the least common denominator for multiple fractions?

Tip: When adding or subtracting fractions, always look for the least common denominator to make the process easier.