Answer:
To make the expressions π76n and π2n rational, we need to find the smallest value of n that will eliminate the irrational component, π.
Let's start with the expression π76n. We want to eliminate π, so we need to find a value of n that will cancel out π. We can do this by setting the expression equal to a rational number, say 0:
π76n = 0
To solve for n, we divide both sides of the equation by π:
76n = 0 / π
76n = 0
Since any number multiplied by 0 is 0, we find that n can be any value, including 0, for the expression π76n to be rational.
Now let's move on to the expression π2n. We want to eliminate π again, so we set the expression equal to a rational number, say 26:
π2n = 26
To solve for n, we divide both sides of the equation by π:
2n = 26 / π
Now, in order for 2n to be rational, 26/π must also be rational. However, π is an irrational number, and dividing any rational number by an irrational number will result in an irrational number. Therefore, there is no value of n that will make the expression π2n rational.
In summary, the smallest value of n that will make each number rational is n = 0 for the expression π76n. However, there is no value of n that will make the expression π2n rational.