billy.zhang

Removing denominators, we get \(5x+5y=2x-2y\). Combining like terms: \(3x=-7y\). After dividing by \(3y\) on both sides, we see that \(\frac xy= -\frac73\)

Rewritten in latex, this is \(9y = 3\cdot 6\cdot 27\cdot \frac4{81\cdot 2}\)

Dividing both sides by \(9\) we get \(y = 3\cdot 6\cdot 3\cdot \frac4{81\cdot 2}\). Now it's just math.

\(6=2\cdot3\), so we can rewrite this as \(y = 3\cdot 2\cdot 3\cdot 3\cdot \frac4{81\cdot 2}\)

The three \(3\)'s can cancel out with the \(81\) to get \(y = 2\cdot\frac4{3\cdot 2}\).

The \(2\) and the \(\frac12\) cancel until we get \(y= \frac43\). 

Using simple ratio basics, we have \(\frac{85.4}{12.2}=\frac{x}{12.5}\). Multiplying by \(\frac{10}{10}\) we get \(\frac{854}{122}=7=\frac{10x}{125}\)

Multiplying \(125\) and dividing by \(10\) gets us \(x=87.5\), thus the wingspan of the plane is \(87.5\) feet.

Note that the inches and feet didn't really matter, since they all cancelled out anyway. 

We want \(\frac{x-3}{x-1}+\frac{ 6}{x-3}\). finding common denominator, we get \(\frac{(x-3)^2+6(x-1)}{(x-3)(x-1)}\)

Using distrubutive property and combining like terms about a billion times later, we get \(\frac{x^2+3}{(x-1)(x-3)}\), thus \(\text{C}\) is the correct answer choice.

Rewritten in latex, this is \((x^2 + 2x + 5 - x - 4)(3x^2 - x - 4 + 2x + 8) \ge 0\).

Let's simplify the things in the parenthases. Doing so, we get \((x^2+x+1)(3x^2+x+4)\ge0.\) These cannot be simplified further.

However, all the signs in this are positive. So for any value of \(x\) inputted in here, it's always \(\ge 0\). So the solutions to this are \(-\infty , or in interval,  \(x\in(-\infty,+\infty)\).

Expanding and simplifying, we get \(\frac{a+c}b+\frac{a+b}c+\frac{b+c}a\).

Since \(a+b+c=0\), we also have \(a+c=-b, \)

\(c+b=-a,\) and \(a+b=-c.\)

Substituing into the simplified expression, we have \(-1+(-1)+(-1)\), which is just \(\boxed{-3.}\) Very simple problem with nice solution.

The easiest way is to prime factorize \(324000\). We find that it's \(2^5\cdot3^4\cdot5^3\).

We can take out some numbers. Simplifying, we get \(2\cdot3\cdot5\sqrt[3]{2^2\cdot3}=\boxed{30\sqrt[3]{12}}\)

This is as far as it simplifies. I don't know what you mean by a fraction.

Based on the polynomial given, we can assume that the degrees of \(p(x)\) and \(q(x)\) is \(4\).

And since we have no other terms like \(x^2\) or \(x^6\), we can assume that  \(p(x)\) is in the form \(x^4+A\)  for an integer \(A\), and that \(q(x)\) is in the form \(x^4+B\)  for any integer \(B\).

Thus, expanding, we get \(x^8+(A+B)x^4+AB\). From this, we know that \(A+B=98\) and \(AB=1\).

Fortunately, we don't need to find \(A\) and \(B\). What we are trying to find is \(2x^4+(A+B)\), which, after we plug in things, we get \(2+98=\boxed{100}\), which is a perfect, round, number to end the problem.

Hello guest,

I think you mean integer. If so, then we can assume that the first digit is \(3.\) So we're looking for is a \(3\)-digit number greater than \(456\) that adds up to \(9\).

The first digit of this 'mystery' number can't be \(4,\) because the two numbers after it have to add up to \(5\) which is impossible in this case.

Thus the first digit must be \(5.\) The next two numbers must add up to \(4,\) which \(0\) and \(4 \) do perfectly. Thus the smallest integer larger than \(3456\) that satisfys this is \(\boxed{3504.}\)