Absolutely, let's solve these rational numbers step-by-step.
### Part (a)
Given: [tex]\(\frac{-2}{3} \times \frac{1}{5} + \frac{2}{3} \times \frac{1}{5}\)[/tex]
1. Identify a common factor in each term:
[tex]\[
\left(\frac{-2}{3}\right) \times \left(\frac{1}{5}\right) + \left(\frac{2}{3}\right) \times \left(\frac{1}{5}\right)
\][/tex]
2. Factor out the common term [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[
\left(\frac{1}{5}\right) \left(\frac{-2}{3} + \frac{2}{3}\right)
\][/tex]
3. Simplify inside the parentheses:
[tex]\[
\frac{-2}{3} + \frac{2}{3} = 0
\][/tex]
4. Multiply:
[tex]\[
\left(\frac{1}{5}\right) \times 0 = 0
\][/tex]
Thus, the result for part (a) is:
[tex]\[
0
\][/tex]
### Part (b)
Given: [tex]\(\frac{3}{7} \times \left(\frac{-2}{5}\right) + \left(\frac{2}{5}\right) \times \left(\frac{1}{3}\right) \times \left(\frac{9}{7}\right)\)[/tex]
1. First calculate the product [tex]\(\frac{3}{7} \times \left(\frac{-2}{5}\right)\)[/tex]:
[tex]\[
\frac{3}{7} \times \left(\frac{-2}{5}\right) = \left(\frac{3 \times -2}{7 \times 5}\right) = \frac{-6}{35}
\][/tex]
2. Next, calculate the product [tex]\(\left(\frac{2}{5}\right) \times \left(\frac{1}{3}\right) \times \left(\frac{9}{7}\right)\)[/tex]:
[tex]\[
\left(\frac{2}{5}\right) \times \left(\frac{1}{3}\right) \times \left(\frac{9}{7}\right) = \frac{2 \times 1 \times 9}{5 \times 3 \times 7} = \frac{18}{105}
\][/tex]
3. Simplify [tex]\(\frac{18}{105}\)[/tex]:
[tex]\[
\frac{18}{105} = \frac{18 \div 3}{105 \div 3} = \frac{6}{35}
\][/tex]
4. Add the two results together:
[tex]\[
\frac{-6}{35} + \frac{6}{35} = \frac{-6 + 6}{35} = \frac{0}{35} = 0
\][/tex]
Thus, the result for part (b) is:
[tex]\[
0
\][/tex]
Combining our results:
[tex]\[
\text{(a) } 0, \text{ (b) } 0
\][/tex]
So, both expressions simplify to [tex]\(0\)[/tex].
