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Thank you @textot ! This was exactly what i needed!

l=large pizza cost

m=medium pizza cost

d=drink cost

l=4d

m=2d

d=d

total money spent=4d+2d+d=7d

money left is $\boxed{30-7d}$

L'Hopital's rule tells us that if the numerator function (let's call that f(x)) and the denominator function (let's call that g(x)) both approach 0 as x approaches n, where n is a constant, and if \(\lim_{x\rightarrow n} (\frac{f'(x)}{g'(x)})\) exists, then \(\lim_{x\rightarrow n} (\frac{f'(x)}{g'(x)})\).

So basically keep taking the derivative of the top and bottom of the fraction until you reach a point where it's not indeterminate or it's undefined (like 4/0).

In this case, you need to use the power rule:

\(\lim_{x\rightarrow 1}\frac{x^3-3x^2+3x-1}{x-1}\\=\lim_{x\rightarrow 1}\frac{3x^2-6x+3}{1}\\ 3(1)^2-6(1)+3=0\)

The digits repeat every 6 digits, so the 2007th digit is equal to the 3rd digit. The 3rd digit is 1, so your answer is $\boxed{1}$

Let $x=0.\overline{515}$.

Multiplying both sides by 1000 gives: $1000x=515.\overline{515}$

Subtract x from 1000x to get rid of the repeating part: $1000x-x=515$ so $999x=515$

Dividing both sides by 999 gives $\boxed{x=0.\overline{515}=515/999}$

$1/5+1/5^2+1/5^3=0.2+0.04+0.008=\boxed{0.248}$

ab = 9*(-30/9) = -30.

There are 22 different non-congruent isosceles triangles.

$0.\overline{42}=42/99=14/33$, so we have $99/100*14/33=\boxed{\frac{21}{50}}$

The probability that he gets 1 digit is 9/12=3/4, and the probability that he gets 2 digits is 3/12 = 1/4, so the expected number of digit he obtains is:

3/4*1+1/4*2=1.25

(a) Let w = a + bi and z = c + di.  The rest is expanding.

(b) Let w = a + bi and z = c + di.  Then

\[\dfrac{w + z}{1 + wz} = \dfrac{a + c + bi + di}{1 + (a + bi)(c + di)}\]

To express this in rectangular form, we can multiply the numerator and denominator by the conjugate:

\[\dfrac{a + c + bi + di}{1 + (a + bi)(c + di)} = \dfrac{(a + c + bi + di)((1 - (a + bi)(c + di))}{(1 + (a + bi)(c + di))(1 - (a + bi)(c + di))}\]

The denominator simplifies to (1 - (a^2 + b^2)(c^2 + d^2)), which is real.  The numerator simplifies to a^2 - b^2 + c^2 - d^2, which is also real.  Therefore, the complex number (z + w)/(zw + 1) is real.

The mystery number is 5/4, and there is no other number satisfying that condition, because:

The denominator must be either 1, 2, 4, 5, or 8, in simplest terms, so that the decimals don't repeat forever, but 1, 2, and 5 cannot work as the denominator, because if you test numbers for the numerator they don't satisfy the condition 1.2

sort of, people post questions, and other people answer them, so I guess it's like reddit for math. The only difference is here you get score for posting questions/answers and having them upvoted/downvoted and whatnot, so it's a bit different.

Thank you @CPhill!

If by $(\overline{345})$ you meant $0.(\overline{345})$, then the answer would be $0.(\overline{123+345})=0.(\overline{468})$. This means the sum of the digits in one repeating period is $4+6+8=\boxed{18}$

It says the solution is incorrect, but it won't give me the full answer. Can you show me all your work so I can look for mistakes?

z^4 + z^2 + 1 is a factor of z^24 - 1, so the answer is n = 24.

$(0.\overline{6})(0.\overline{06})=(6/9)*(6/99)=4/99=0.\overline{04}$

The sum of the first $15$ digits will be $0+4+0+4+0+4...+4+0$. This has length $15$, with $8$ zeroes and $7$ fours. So it is equal to $8(0)+7(4)=\boxed{28}$

The area of a rhombus is given by $\frac{pq}{2}$, where p and q are the diagonals. Plugging in gives $\frac{18*12}{2}=\boxed{108}$

The equation \(x^2+y^2+z^2=1\) represents a sphere with a center at (0, 0, 0) and a radius of 1, and the volume of the space that contains all the possible points of \((x, y, z)\) is a cube with a side length of 2. The volume of the sphere is \(\frac{4}{3}\pi\), and the volume of the cube is \(8\), so the probability that \(x^2+y^2+z^2\leq1\) is \(\frac{\frac{4}{3}\pi}{8}=\boxed{\frac{\pi}{6}}\)

Cheerios$=99$ cents

Tasteeos$=8/9$(Cheerios)$=8/9(99)=8*11=\boxed{89}$ cents

Just listing out the terms, we get:

1/2+1/3=5/6

1/2+1/3+1/4=13/12

1/2+1/3+1/4+1/5=77/60

1/2+1/3+1/4+1/5+1/6=87/60

1/2+1/3+1/4+1/5+1/6+1/7=223/140

1/2+1/3+1/4+1/5+1/6+1/7+1/8=481/280

1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9=4609/2520

1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10=4861/2520

1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11=55991/27720(first number greater than 2)

From this, we can see the smallest integer value of x is 11. However, this way of listing out terms is slow, and if anybody else has an algebraic way, it would be much appreciated.

Es ist okay, ich habe es verstanden (idk i am using google translate)