We are given an equation that shows the number of possible random walks w.
w= d^nHere, n is the number of steps you take in a d-direction random walk. We are asked to find the number of steps taken in a 2-direction random walk if there are 4096 possible walks. To do so, we will substitute 2 for d, and 4096 for w in the equation.
By the Power Property of Equality, the exponents are equal to each other.
2^()darkviolet12=2^n ⇔ 12= n
Therefore, 12 steps have been taken in a 2-direction random walk if there are 4096 possible walks.
b
This time we need to substitute 4 for d, and 65 536 for w in the equation.
By the Power Property of Equality, the exponents are equal to each other.
4^()darkviolet8=2^n ⇔ 8= n
Therefore, 8 steps have been taken in a 4-direction random walk if there are 65 536 possible walks.
c
For this part, we are asked to find the number of directions if a walk of 7 steps has 2187 possible walks. To find it, we will substitute 7 for n and 2187 for w in the equation.
In order for this equality to hold, the bases should be equal because of the Power Property of Equality.
3^7= d^7 ⇔ 3= d
Therefore, 3 directions could be taken at each step.
