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sum_(k=0)^10 3 (k+2) 2^k = 67,584

Who am talking to?    You should get an account and then you could private message me. :)

At present I am listening to this:

melody what are you doing today?

hi,IT'S MY BIRTHDAY EVERYONE YAY!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

dragonslayer554 nd everybody it's my birthday! By the way I'm Alice.

Mmm 

Let x = 1-0.02  This is unnecessary but just for easier understanding.

\(\text{Original expression}\\ =\dfrac{2x^{10}}{x}\\ =2x^9\\ \text{Now back-substitute the values.}\\ =2(1-0.02)^9\\ =2\cdot(0.98)^9\\ =2\cdot0.83374776213\\ =1.66749552426\)

Welcome back, Guest! :D

\(\begin{array}{rll}3^{81x}&=&81^{3x}\\ 81x\cdot \log_33 &=& 3x\cdot\log_381\\ 81x&=&12x\end{array}\\ \mbox{The only solution is }x=0\)

2 + 2 = 4, isn't it?

Example:

2.653 x 3.332

estimated: 3 x 3 = 9

actual: 2.653 x 3.332 = 8.839796

Close enough, isn't it?

Multiply 100 to both the numerator and denominator.

\(\begin{array}{rl}&\dfrac{1}{1.01}\\=&\dfrac{100}{101}\end{array}\)

Exponents smaller than whole number, I bet you mean fractions and negative numbers.

\(\boxed{\color{red}m^{\frac{a}{n}}=\sqrt[n]{m^a}}\)

\(\boxed{\color{red}m^{-n}=\dfrac{1}{m^n}}\)

Melody, 179 x 3 x 5 = 2685 is smaller and fits in the requirement.

https://www.desmos.com/calculator/ddksm4r2rw

Me too Guest #1, These numbers doesn't make sense.

what are the roots of y=2x times 2x-7x-4

Do you meant the roots of

\(y=2x\times(2x-7x-4)\\ y=2x\times(-5x-4)\\\)

Roots are when y=0

so  2x=0     or      -5x-4=0

        x=0     or         x=-4/5

Guest 2, yours does not work because 9 is not a prime number.

The prime factorization of 1995, which is 3*5*7*19, uses each odd digit exactly one and 1995 is the smallest positive integer with this property. What is the next smallest?

Please show steps for solving. 

1 and 9 are the only digits that is not prime so they have to be used in conjuction with  digits

Mmm

maybe   3*5*7*91

No, 91 is not prime so that is not any good :(

So I want the smallest 3 digit number containing 9,1 and either 3,5,or 7

mabe   139*5*7

139 is prime  :)

So I am going with 139*5*7 = 4865

5 - 2 = 3.

I think I found something:

If Log (A) = x then A = 10^x right?

Then can we say 

log ( (1000  ( square root ( ( 10^x)^3 ) ) ) / ( ( 10^x) ^-1) )      is = to the original equation???

Only rule was to find an algebric expression containing only "x" as a variable that would be equal to the original one.

Nevermind. Doesn't work.

For 4 integers (w, x, y, z), it is ((((w+x+y+z)/4)-((abs(z-y-x-w))/4))*(((w+x+y+z)/4)+((abs(z-y-x-w))/4))). I think (not 100% sure).

3*5*7*19=1995. The factors have 1, 3, 5, 7, and 9 in them (the first 5 odd numbers). Since you want to find the next lowest number with this property, you have two options. When you multiply two positive integers (say x and y), what you are doing is ((((x+y)/2)-((abs(y-x))/2))*(((x+y)/2)+((abs(y-x))/2))), or ((((x+y)/2)^2)-(abs(y/2-x/2))). So the farther apart x and y are, the lower answer you get (10*10=100, 11*9=99, 10+10=20, 11+9=20, abs(10-10)=0, abs(11-9)=2). So the options are 1*5*7*39, or 3*5*9*17. The first option is problematic because 1 isn't prime, so it would have to be 7*15*39. 3*5*9*17=2295, while 7*15*39=4095. If you try to improve on this, it eventually leads back to 3*5*9*17. So the answer is 2295.

Find the inverse of:  f(x)  =  log₂(4x) - 2.

To make the notation simpler, replace  f(x)  with y:

     y  =   log₂(4x) - 2

Interchange  'x'  and  'y':

     x  =  log₂(4y) - 2

Solve for y:

     x + 2  =  log₂(4y)

which is:

 log₂(4y)  =  x + 2

Write in exponential form:

        4y  =  2^{x + 2}

          y  =  (2^{x + 2}) / 4

--->    y  =  (2^{x + 2}) / 2²

--->    y  =  2^x

So, the inverse function is:  y(x)  =  2^x