Express $156$ as a product of its prime factors.

Without actually performing the long division,state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion: $\frac{129}{2^{2} \cdot 5^{7} \cdot 7^{5}}$

Express $140$ as a product of its prime factors.

Express $3825$ as a product of its prime factors.

Find the $LCM$ and $HCF$ of the following pairs of integers and verify that $LCM \times HCF =$ product of the two numbers $336$ and $54$.

Prove that $\frac{1}{\sqrt{2}}$ is an irrational number.