azsun

The number 12 can be broken up to $x^2*y^2$, since the number of factors is $(2+1)(2+1)=3\cdot3=9.$ The three numbers do not work here since there are no possible outcomes.

Thus, we can have: $2^2*3^2=36.$

$2^2*5^2=100.$

$2^2*7^2=196.$

I do not think any more possible outcomes are there, and you can still check my work.

Thus, the answer is $36+100+196=\boxed{332}.$

$8=2^3.$

Thus, we have $\left \lfloor \frac{100}{2} \right \rfloor+\left \lfloor \frac{100}{4} \right \rfloor+...+\left \lfloor \frac{100}{32} \right \rfloor+\left \lfloor \frac{100}{64} \right \rfloor=97$. Since $97=32*3+1$, $\boxed{32}$ is our searched answer.

-azsun

We can also represent these using combinations.

We have to find the value of $\binom{10}{8}.$

A quicker way is to recognize Pascal's triangle, and this is the tenth row, and the sum is $2^{10}=1024.$ Thus, the answer is, $\frac{\binom{10}{8}}{1024}=\boxed{\frac{45}{1024}}.$ 

I think it's x=-4, y=1. 

Try to rewrite the expression as the sum of squares plus a number.

Is the answer -11? Inspiration: https://www.quora.com/If-x-and-y-are-real-numbers-then-what-is-the-minimum-value-of-x-2-+4xy+6y-2-4y+4

Don't take my word for it, though. 

Solving these equations gives us: x=y*tan(29), and x=(y+100)(tan25).

Use a unit circle to find the appropriate values. y should be approximately 529.9. Then, solve for x by plugging the value of y back in.

$2.8-6=\boxed{-3.2}$.

We first go by the exterior angle measure: $\frac{360}{n}$.

Now, $n$ has to be a factor of $360$ , so we count how many factors this number has.

Since $360=2^3*3^2*5$ , we add one to each power, resulting in    $4*3*2=24$ factors.

Finally, a polygon cannot have one or two sides; thus the answer is $24-2=\boxed{22}$ regular polygons.

Since they are walking in opposite directions, we add their rates. 3+4=7mph. Now, we divide from 14 to get an answer of 2.

Good solution, but it's $5*5,$ not $5.5$ .

Hey, Manuel! What's up?

Thank you, everyone! 

Wow! I wasn't active in such a long time, I'm so sorry. Finally, I saw this post! Happy birthday CPhill, hope you have a blast!

Enjoy my recipe:

Oreo flavor!

That seems better! Thank you, CPhill! 

Translating this, we get $\ln \left(10^{z+2}\right)=\ln \left(27\right).$ Now, we just apply the log rule($\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)$) , and get  $\left(z+2\right)\ln \left(10\right)=\ln \left(27\right).$

Then, we can simplify this to be  $\left(z+2\right)\ln \left(10\right)=3\ln \left(3\right).$So, we solve it, and our final answer is $z=\frac{3\ln \left(3\right)}{\ln \left(10\right)}-2.$

Continuing on, we have $\frac{24}{2}, \frac{-20}{2}$. Thus, our answer is $(12,-10)$.

I hope Rom will give you a detailed answer, but there are 10 possible choices out of 90 values, so 10/90=1/9. 

For Question 3, I'm getting two different answers. 9, is my first take.

Hint: Difference of Squares!
a^2+b^2=c^2, c^2-b^2=a^2, (c+b)(c-b)=36.

You also know the perimeter is 18, so 6+b+c=18, b+c=12.

So, (12)(c-b)=36, c-b=3..

Now, we have two equations, and if we subtract, we will get: c=7.5