CubeyThePenguin

mass = 0.25 kg

initial velocity = v_i = 6 m/s

coefficient of kinetic friction = mu_k = 0.1

final velocity = v_f = 0 m/s

distance = d = ?

There are three forces acting on the puck (friction, gravity, normal force) Gravity and normal force cancel each other out, so we can ignore that for now. 

Friction acts in the -x direction.

F_n = mg = (0.25 kg)(9.8 m/s^2) = 2.45 N

F_f = (mu_k) * (F_n) = (0.1) * (2.45) = 0.245 N

F = ma    ==>    a = F/ m = (0.245 N) / (0.25 kg) = 0.98 m/s^2

Finally, we use kinematics.

(v_f)^2 = (v_i)^2 + 2ad

0 = (6^2) + 2(0.98)d

d = 18.367 m

My physics is a bit rusty so this is the best I can come up with.

x = 2 ==> every value of x^y is positive             {1/4, 1/2, 1, 2, 4}

x = 1 ==> every value of x^y is positive              {1}

x = -1 ==> x^y is positive if y is even                {-1, 1}

There are 5 distinct values of x^y: 1/4, 1/2, 1, 2, 4

From what you typed, maybe this is what you mean.

\(\sqrt[3]{8x} = 2 \sqrt[3]{x}\)

Use the Pythagorean Theorem.

This is an exponential function

\(2^x = 50\)

\(x = \log_2{50} = 5.643\)

It takes 5 stages to notify all the players.

Oops, my bad. Your answer is correct.

\((8 \%)(75) = \frac{8}{100}(75) = \boxed{6}\) 3 pointers

Count the probability that she delivers all the letters incorrectly. This is the number of derangements of 5, which is 44.

Probability: 1 - 44/120 = 19/30

Use complementary counting and find the number of boxes that don't  have a bobblehead inside.

\(1 - (\frac{1}{6})^3 = \boxed{\frac{215}{216}}\)

English translation: What is 17 parts 30 plus 3 parts 20 times 2 ?????

\(\frac{17}{30} + \frac{3}{20} \cdot 2\)

\(\frac{34}{60} + \frac{9}{60} \cdot 2 = \frac{52}{60} = \boxed{\frac{13}{15}}\)

**edit: fixed