The numerator (top number of a fraction) is the total number of parts we have. The denominator (bottom number of a fraction) is the number of parts something is divided into.

The rope is first cut in half, so there are 2 pieces.

Then, those 2 pieces are cut in half. Now, there are 4 pieces.

Finally, those 4 pieces are cut in half. Now, there are 8 pieces.

The rope is divided into 8 total pieces, so the denominator is 8. The question asks about each piece, which means the numerator is 1. That makes the fraction 1/8. Each piece of the rope is 1/8 of the original length of the rope.