How can astronomers infer approximately how long the universe has been expanding?

Not only astronomers, but all scientists use the technique of rock dating to determine the age of the universe. From the Big Bang Theory, all the celestial bodies were made at the same time. So, the age of the Earth is just the same with the age of the universe. So, they gather rocks deep down in the crust of the Earth and let it undergo radioactive decay. By determining the amount of radioactive material left and half-life, they could compute the age of the Earth.