In a tennis competition, 128 players were selected for a series of knockout rounds. In each round the losers were eliminated and the winners reached the next round. How many players moved to the next round after 4th round? Express this number in the exponential notation in terms of the initial number of players.

Given:

Initial number of players = 128.

In a knockout round, the number of players is reduced to half $\Big(\dfrac{1}{2}\Big)$.

After 1st round, players left = $\dfrac{1}{2} \times 128$.

After 2nd round, players left = $\dfrac{1}{2} \times \dfrac{1}{2} \times 128 = \Big(\dfrac{1}{2}\Big)^2 \times 128$.

After 3rd round, players left = $\Big(\dfrac{1}{2}\Big)^3 \times 128$.

After 4th round, players left = $\Big(\dfrac{1}{2}\Big)^4 \times 128$ $= \dfrac{1^4}{2^4} \times 128 \\[1em] = \dfrac{128}{2^4} \\[1em] = \dfrac{128}{2 \times 2 \times 2 \times 2} \\[1em] = \dfrac{128}{16} \\[1em] = 8$

Hence, 8 players moved to the next round.

Exponential notation in initial no. of players = $\dfrac{\bold{128}}{\bold{2}^\bold{4}}$