By looking for the oldest stars. The oldest stars in the universe are thought to be about 13.6 billion years old. This is because stars form from gas clouds, and the gas clouds that formed the oldest stars were only able to collapse and form stars after the Big Bang had cooled down enough.
By measuring the rate of expansion of the universe. The universe is expanding, and the rate of expansion is accelerating. This means that the universe was smaller in the past, and therefore the Big Bang must have happened more recently. The current best estimate for the age of the universe, based on this method, is 13.8 billion years.
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