Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\bf \:(1) \: \: - \: \dfrac{40}{49} \qquad \: \\ \\& \qquad \:\bf \: (2) \: \: \: \: \: \: \: 6 \dfrac{4}{11} \end{aligned}} \qquad \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given expression is
[tex]\qquad\sf \: - 1\dfrac{1}{7} \times \dfrac{5}{7} \\ \\ [/tex]
[tex]\qquad\sf \: = \: - \: \dfrac{7 \times 1 + 1}{7} \times \dfrac{5}{7} \\ \\ [/tex]
[tex]\qquad\sf \: = \: - \: \dfrac{7 + 1}{7} \times \dfrac{5}{7} \\ \\ [/tex]
[tex]\qquad\sf \: = \: - \: \dfrac{8}{7} \times \dfrac{5}{7} \\ \\ [/tex]
[tex]\qquad\sf \: = \: - \: \dfrac{40}{49} \\ \\ [/tex]
Hence,
[tex]\qquad\bf \: - 1\dfrac{1}{7} \times \dfrac{5}{7} \: = - \:\dfrac{40}{49} \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
Given expression is
[tex]\qquad\sf \: - 1\dfrac{1}{9} \times \left( - 5\dfrac{8}{11}\right) \\ \\ [/tex]
[tex]\qquad\sf \: = \: 1\dfrac{1}{9} \times 5\dfrac{8}{11} \\ \\ [/tex]
[tex]\qquad\sf \: = \: \dfrac{9 \times 1 + 1}{9} \times \dfrac{11 \times 5 + 8}{11} \\ \\ [/tex]
[tex]\qquad\sf \: = \: \dfrac{9 + 1}{9} \times \dfrac{55 + 8}{11} \\ \\ [/tex]
[tex]\qquad\sf \: = \: \dfrac{10}{9} \times \dfrac{63}{11} \\ \\ [/tex]
[tex]\qquad\sf \: = \: \dfrac{10}{1} \times \dfrac{7}{11} \\ \\ [/tex]
[tex]\qquad\sf \: = \: \dfrac{70}{11} \\ \\ [/tex]
[tex]\qquad\sf \: = \: 6\dfrac{4}{11} \\ \\ [/tex]
Hence,
[tex]\qquad\sf\implies \bf \: - 1\dfrac{1}{9} \times \left( - 5\dfrac{8}{11}\right) = 6\dfrac{4}{11} \\ \\ [/tex]