CubeyThePenguin

First, find the total number of ways to get from (0, 0) to (6, 7). There are 6 horizontal steps and 7 vertical steps for a total of 13 steps, so there are \(\binom{13}{6} = 1716\) ways.

Then, find the number of paths that pass through (4, 3). From (0, 0) to (4, 3), there are 4 horizontal steps and 3 vertical steps for a total of 7 steps. \(\binom{7}{4} = 35\)

We then calculate the number of paths from (4, 3) to (6, 7). There are 2 horizontal steps and 4 vertical steps for a total of 6 steps, giving \(\binom{6}{2} = 15\). Overall, there are 35 * 15 = 525 paths that pass through (4, 3), where the frog is. We don't want that, so we subtract this number from the total number of paths to get the answer.

We want to find the smallest n such that \(\binom{n}{3} > 365.\)

\(\frac{(n)(n-1)(n-2)}{6} > 365\)

\((n)(n^2-3n+2) > 2190\)

\(n^3 - 3n^2 + 2n - 2190 > 0\)

n has to be 15 or greater, so the smallest positive integer n that works is 15.

1) \(1 \frac{2}{3} \cdot 1 \frac{1}{4} = \frac{5}{3} \cdot \frac{5}{4} = \boxed{\frac{25}{12} = 2 \frac{1}{12}}\)

2) \(N \cdot \frac{3}{8} = \frac{5}{3}\)

To isolate N, multiply both sides by the reciprical of 3/8, which is 8/3.

\(N = \frac{5}{3} \cdot \frac{8}{3} = \boxed{\frac{40}{9}}\)

3) \(2 \frac{3}{4} \div \frac{1}{8} = 2 \frac{3}{4} \cdot 8 = \frac{11}{4} \cdot 8 = \boxed{22}\)

4) \(\text{cashew nuts mass}= \frac{2}{5} \cdot 2 \frac{1}{2} = \frac{2}{5} \cdot \frac{5}{2} = 1\)

\(\text{total mass} = 2 \frac{1}{2} + 1 = 3 \frac{1}{2} = \frac{7}{2}\)

\(\text{mass per packet} = \frac{7/2}{8} = \boxed{\frac{7}{16}}\)

\(\sqrt{64} = \sqrt{8^2} = 8^{2 \cdot \frac{1}{2}} = \boxed{8}\)

These answers are equivalent, as shown below:

\(\frac{15}{4 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \boxed{\frac{15\sqrt{2}}{8}}\)

This process is called rationalizing the denominator.

\(\frac{30 + 19i}{2+5i} = \frac{30 + 19i}{2+5i} \cdot \frac{2-5i}{2-5i} = \frac{60+38i-150i+95}{4+25} = \boxed{\frac{155}{29} - \frac{112i}{29}}\)

c = 4d

b = 2c = 8d

a = 3b = 24d

(a + c)/(b + d) = (28d)/(9d) = 28/9

\(\frac{1}{\sqrt{2}} + \frac{3}{\sqrt{8}}+\frac{5}{\sqrt{32}}\)

\(= \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} + \frac{3}{2\sqrt{2}} + \frac{5}{4\sqrt{2}}\)

\(= \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{3}{2 \sqrt{2}} + \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{5}{4\sqrt{2}}\)

\(= \frac{\sqrt{2}}{2} + \frac{3 \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8}\)

\(= \frac{4 \sqrt{2} + 6\sqrt{2} + 5\sqrt{2}}{8}\)

\(=\boxed{\frac{15\sqrt{2}}{8}}\)

1 - 0.87 = 0.13

Find the discriminant (b^2 - 4ac) and see which equation has a discriminant of 0.

4x^2 + 16x + 8 ---> 16^2 - 4(4)(8) = 128

-x^2 + 4x + 5 ---> 4^2 - 4(-1)(5) = 36

9x^2 - 6x + 1 ---> (-6)^2 - 4(9)(1) = 0

We know that only one quadratic has one distinct root, so we can stop checking here.

9x^2 - 6x + 1 = 0

(3x - 1)^2 = 0

3x - 1 = 0

x = 1/3