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"Suppose a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,where a,b,c are integers whose greatest common divisor is 1. Determine |c|."

More importantly members have their questions saved in a watchlist and they can, if they chose, to be notified if an answer comes in.

hectictar: I actually knew quite some time ago that your "username" was a anagram of  what you wanted to become when you finished your school! I knew that from your beautiful drawings. You will make a great "Architect"!,

If you don't require the FULL computation, you can use "Stirling's Approximation Formula":

S =n! ~ sqrt(2.pi.n) * (n/e)^n =Sqrt(2 x 3.141592 x 386) x (386 / 2.718281828)^386 = 

3.0020171928170054524994166439502 x 10^832

386! = 300 2665366520 8755582903 3345000063 0933184096 3550392751 1664147182 0499734725 8296126846 5252516059 9730397713 6252641384 5705669545 1577926013 1893735727 4065594295 4866195550 0233594014 1755020797 2844581987 7971506520 8866445606 3529812510 5856946586 3222721808 4121192521 5135767764 3686166365 8503648896 5559254328 3996428349 7349348049 1331159927 0875024868 6527579059 3419975694 0012558821 7002911036 2048989780 4985303473 6850964460 3348329817 4632206323 9807745300 6125177810 2438021524 4451216066 6763938734 9877224771 2704913550 1839059825 8287209819 2708586029 1680960180 6857568765 4265894797 4099940060 0414826937 0397070910 6242919910 2694366291 7872935014 6346432168 1045878558 9853876791 7170874046 9523706315 0045348787 4232848718 4228390421 8581564848 1526066812 2293859261 6579736545 5257600000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 =833 digits long.

Here is the first equation:

I'll leave you to use the same technique on the second equation.

 Please help! Do not know hot to solve

\(\begin{array}{|rcl|rcl|} \hline g(x) &&& f(x)&=&g^{-1}(x) \\ \hline g(-15)&=&0 & f(0)&=&-15 \\ g(0)&=&3 & f(3) &=& 0 \\ g(3)&=&9 & f(9) &=& 3 \\ g(9)&=&20 & f(20) &=& 9 \\ \hline \end{array} \)

\(\mathbf{f(f(9))} = f(3) \mathbf{= 0} \)

the water park is a popular field trip destination. falls high filled 2 vans and 9 buses with 383 students.

rainy high filled 8 vans and 3 buses with 179 students. find the number of students in each van and each bus

\(\text{Let $\mathbf{v} =$ students in a $\mathbf{\text{van} }$ } \\ \text{Let $\mathbf{b} =$ students in a $\mathbf{\text{bus} }$ } \)

\(\begin{array}{|lrcll|} \hline \text{rainy high}: & 8v + 3b &=& 179 \quad | \quad \cdot 3 \\ & 24v + 9b &=& 537 \quad (1) \\\\ \text{falls high}: & 2v + 9b &=& 383 \quad (2) \\ \hline (1)-(2): & 24v + 9b - ( 2v + 9b ) &=& 537 - 383 \\ & 24v + 9b - 2v + 9b &=& 154 \\ & 22v &=& 154 \\ & v &=& \frac{154}{22} \\ & \mathbf{ v } &=& \mathbf{7} \\\\ & 2v + 9b &=& 383 \\ & 2\cdot 7 + 9b &=& 383 \\ & 14+ 9b &=& 383 \\ & 9b &=& 369 \\ & b &=& \frac{369}{9} \\ & \mathbf{ b } &=& \mathbf{41} \\ \hline \end{array}\)

 In each van are 7 students.
 In each bus are 41 students.

I like that you know members have priority yet still won't make an account to have priority as well.

lol this guest

because members have priority over guests like me

a)   We could use the Law of Cosines to find  θ , but since  △ABC  is an isosceles triangle, we can split it into two

      congruent right triangles by drawing a height from  C  to side AB, like this:

b)   arc length   =   (radius) * (angle measured in radians)   ≈   (20 cm) * (1.287)   ≈   26 cm

c)

       i)   sector area   =   (angle measured in radians)/2 * (radius)²  ≈   (1.287)/2 * (20)²  cm²   ≈   257.4  cm²

       ii)  shaded area   =   sector area - triangle area     ...Do you know how to find the area of the triangle?

I mean, I don't think it is a problem, but I think it would be better for them if they made an account.

Yep! THX hectictar! and why is your username hectic-tar..... tar isn't very hectic in my experience...

Thank you! but you found the volume, and you added.... so thx for trying! The question asks for the surface area and subtraction tho...

1) one is kinda easy..... just start dividing and find a pattern. I found the pattern 0.370 repeating, and so your answer for 1 would be 0.3703703703

2)So I started manually dividing and after 6 digits, I stopped... I want to show it here but have no idea how... :( but your answer is 923076

3)to convert 0.045 repeating 5 to fraction form, just do this:

       1000x=45.55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555

             - x=0.045555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555

              _____________________________________________________________________________________________________

subtract and get 999x=45.1

                                 x=451/9990

                                 x=41/90

4) \(6032_8\) to decimal?! argh! ok, so start by doing (6*6^3)+(0*6^2)+(3*6^1)+(2*6^0)=1407.... and 1407 is 6032_8 in base 10.... in other words, decimal.

5)\(999__16\)ok I cant get the base thing to work in latex... so deal with this.

(9*16^2)+(9*16)+9=2457 in decimal..

6)35/2=17r1

  17/2=8r1

   8/2=4r0

   4/2=2r0

   2/2=1r0

   1/2=0r1

so i think 35 in base 2 is 100011

7)10101101...

That wouldn't be divisible by 3, but there would always be another combination of 3 numbers in the set that would be divisible by 3.

OK, IM guessing this is from AoPS from the dollar signs...... I don't really mind the asking and I would solve it but can you edit it so the latex is in latex???????? ok so this is my version of it...:

1.) Find the first ten digits after the decimal point in the decimal expansion of \(\frac{10}{27}=0.abcdefghij\ldots\)

 without a calculator.

(Express your answer as a ten-digit number.)

2.) The number \(\frac{12}{13}\) can be expressed as a repeating decimal \(0.\overline{abcdef}\). Find the repeating part \(abcdef\) without a calculator.

3.) Convert \(0.04\overline{55}\) to a fraction in simplest form.

4.) Convert \(6032_8\) to decimal.

5.) Convert \(999_{16}\) to decimal.

6.) Convert 35 from base 10 to base 2.

(You do not need to include the subscript 2 in this answer.)

7.) Convert \(2231_4\) to base 2 without first converting to decimal.

Yes they do Nickolas, non linear just do not graph as lines, they graph as other things. Like curves that are not straight!