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**Temas: Operaciones con Números Racionales y Polinomios Aritméticos**

1. Determinar la adición de los números racionales representados gráficamente.

2. Resuelve las operaciones:

\[
\begin{array}{ll}
\frac{3}{7}+\frac{2}{5} & \frac{9}{7}-\frac{1}{21} \\
-\frac{1}{8}+\frac{5}{6} & \frac{7}{30}-\frac{2}{45} \\
\frac{9}{16}+\left(-\frac{11}{64}\right) & -\frac{8}{27}-\frac{8}{81} \\
\left(-\frac{7}{40}\right)+\frac{7}{16} & \frac{3}{32}-\left(-\frac{1}{48}\right) \\
\left(-\frac{2}{15}\right)+\left(-\frac{1}{12}\right) & -\frac{1}{28}-\left(-\frac{1}{42}\right)
\end{array}
\]

3. Escribe "F" si es falso y "V" si es verdadero:

- En la multiplicación de números racionales, el orden de los factores no afecta el producto.
- El inverso multiplicativo de \(\frac{38}{45}\) es \(-\frac{38}{45}\).

Answer :

Sure! Let's solve the problems step-by-step.

### Problem 2: Resolve the operations

We need to perform operations with rational numbers. A rational number is simply a fraction where both the numerator and the denominator are integers.

Here are the operations given:

1. [tex]\( \frac{3}{7} + \frac{2}{5} \)[/tex]: To add these fractions, we need a common denominator. The least common denominator (LCD) of 7 and 5 is 35.
- Convert [tex]\(\frac{3}{7}\)[/tex] to [tex]\(\frac{15}{35}\)[/tex]
- Convert [tex]\(\frac{2}{5}\)[/tex] to [tex]\(\frac{14}{35}\)[/tex]
- Add them: [tex]\( \frac{15}{35} + \frac{14}{35} = \frac{29}{35} \)[/tex]

2. [tex]\( \frac{9}{7} - \frac{1}{21} \)[/tex]: The LCD here is 21.
- Convert [tex]\(\frac{9}{7}\)[/tex] to [tex]\(\frac{27}{21}\)[/tex]
- Subtract: [tex]\( \frac{27}{21} - \frac{1}{21} = \frac{26}{21} \)[/tex]

3. [tex]\( -\frac{1}{8} + \frac{5}{6} \)[/tex]: The LCD is 24.
- Convert [tex]\(-\frac{1}{8}\)[/tex] to [tex]\(-\frac{3}{24}\)[/tex]
- Convert [tex]\(\frac{5}{6}\)[/tex] to [tex]\(\frac{20}{24}\)[/tex]
- Add them: [tex]\( -\frac{3}{24} + \frac{20}{24} = \frac{17}{24} \)[/tex]

4. [tex]\( \frac{7}{30} - \frac{2}{45} \)[/tex]: The LCD is 90.
- Convert [tex]\(\frac{7}{30}\)[/tex] to [tex]\(\frac{21}{90}\)[/tex]
- Convert [tex]\(\frac{2}{45}\)[/tex] to [tex]\(\frac{4}{90}\)[/tex]
- Subtract: [tex]\( \frac{21}{90} - \frac{4}{90} = \frac{17}{90} \)[/tex]

5. [tex]\( \frac{9}{16} + \left(-\frac{11}{64}\right) \)[/tex]: The LCD is 64.
- Convert [tex]\(\frac{9}{16}\)[/tex] to [tex]\(\frac{36}{64}\)[/tex]
- Add: [tex]\( \frac{36}{64} - \frac{11}{64} = \frac{25}{64} \)[/tex]

6. [tex]\( -\frac{8}{27} - \frac{8}{81} \)[/tex]: The LCD is 81.
- Convert [tex]\(-\frac{8}{27}\)[/tex] to [tex]\(-\frac{24}{81}\)[/tex]
- Subtract: [tex]\( -\frac{24}{81} - \frac{8}{81} = -\frac{32}{81} \)[/tex]

7. [tex]\(\left(-\frac{7}{40}\right) + \frac{7}{16} \)[/tex]: The LCD is 80.
- Convert [tex]\(-\frac{7}{40}\)[/tex] to [tex]\(-\frac{14}{80}\)[/tex]
- Convert [tex]\(\frac{7}{16}\)[/tex] to [tex]\(\frac{35}{80}\)[/tex]
- Add: [tex]\( -\frac{14}{80} + \frac{35}{80} = \frac{21}{80} \)[/tex]

8. [tex]\( \frac{3}{32} - \left(-\frac{1}{48}\right) \)[/tex]: The LCD is 96.
- Convert [tex]\(\frac{3}{32}\)[/tex] to [tex]\(\frac{9}{96}\)[/tex]
- Convert [tex]\(-\frac{1}{48}\)[/tex] to [tex]\(-\frac{2}{96}\)[/tex]
- Subtract: [tex]\( \frac{9}{96} + \frac{2}{96} = \frac{11}{96} \)[/tex]

9. [tex]\(\left(-\frac{2}{15}\right) + \left(-\frac{1}{12}\right) \)[/tex]: The LCD is 60.
- Convert [tex]\(-\frac{2}{15}\)[/tex] to [tex]\(-\frac{8}{60}\)[/tex]
- Convert [tex]\(-\frac{1}{12}\)[/tex] to [tex]\(-\frac{5}{60}\)[/tex]
- Add: [tex]\( -\frac{8}{60} - \frac{5}{60} = -\frac{13}{60} \)[/tex]

10. [tex]\(-\frac{1}{28} - \left(-\frac{1}{42}\right) \)[/tex]: The LCD is 84.
- Convert [tex]\(-\frac{1}{28}\)[/tex] to [tex]\(-\frac{3}{84}\)[/tex]
- Convert [tex]\(-\frac{1}{42}\)[/tex] to [tex]\(-\frac{2}{84}\)[/tex]
- Subtract: [tex]\( -\frac{3}{84} + \frac{2}{84} = -\frac{1}{84} \)[/tex]

### Problem 3: True or False Statements

1. _"In multiplication of rational numbers, the order of factors does not affect the product."_
This statement is True because multiplication of numbers is commutative, meaning the order can be changed without affecting the result.

2. _"The multiplicative inverse of [tex]\(\frac{38}{45}\)[/tex] is [tex]\(-\frac{38}{45}\)[/tex]."_
This statement is False. The multiplicative inverse of a number [tex]\( \frac{a}{b} \)[/tex] is [tex]\( \frac{b}{a} \)[/tex], not the negative of the number, so it should be [tex]\(\frac{45}{38}\)[/tex].

I hope this helps! Feel free to reach out if you have more questions.

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