Melody

2xy-4ac+5yx+4ac

Since x*y   is the same as   y*x    2xy and 5yx are like terms and you can add them together to get 7xy  (or 7yx)

2xy-4ac+5yx+4ac

=2xy+5yx+4ac-4ac

=2xy+5xy+0

=7xy

If f(x)+x^5[f(x)]^3=2052 and f(2)=4, find f′(2)

\(f(x)+x^5[f(x)]^3=2052\qquad f(2)=4,\qquad find f′(2)\\ f'(x)+x^5*3[f(x)]^2*f'(x)+5x^4*[f(x)]^3=0\\ f'(2)+2^5*3[f(2)]^2*f'(2)+5*2^4*[f(2)]^3=0\\ f'(2)+32*3[4]^2*f'(2)+5*16*[4]^3=0\\ f'(2)+32*3*16*f'(2)+5*16*64=0\\ f'(2)+1536*f'(2)+5120=0\\ 1537*f'(2)=-5120\\ f'(2)=\frac{-5120}{1537}\\ \)

Thanks Alan,

I shall take a very good look at your answer :)

Thanks Alan,

I never doubted you but I'd still like to see it done in a more general way. ://

(7/4) (21^2) <tan (180/7)^-1 What is this answer?  

I am not really sure what your question is

(7/4) (21^2) tan^-1(180/7)

=(7/4)*(21^2)*atan (180/7) = 67738.7765028982455

or (the second one makes more sense.)

(7/4) (21^2) (tan (180/7))^-1

=(7/4)*(21^2)/(tan (180/7)) = 1602.5553878031321963

Cinema tickets costs 110 $ for adults and 90 $ for kids. They sold 120 tickets for a total of 11640 $. How many adults saw the show?

K+A=120

so   K=120-A

110A+90K=11640

divide both sides by 10

11A+9K=1164

Sub in K=120-A

11A+9(120-A)=1164

11A+1080-9A=1164

2A=84

A=42

42 adults saw the show :)

It is really great to see you back Heureka :))

Yes, I think that Ginger is correct.  Does your book's answer agree with Ginger's answer ?

I answered on the original question.

and yes EP is correct :))

Consider the geometric series \(4+\frac{12}{a}+\frac{36}{a^2}+\cdots\)

 If the sum is a perfect square, what is the smallest possible value of a where a is a positive integer?

This is a GP where  T₁=4     and      r= 3/a

 \(S_{\infty}=\frac{T_1}{1-r}=\frac{4}{1-\frac{3}{a}}=4\div \frac{a-3}{a}=\frac{4a}{a-3}\)

Now this sum is a perfect square so the square root of it is a rational number so

let's look at this.

\(\sqrt{\frac{4a}{a-3}}\\ =\frac{\sqrt{4a}}{\sqrt{a-3}}\\ =\frac{2\sqrt{a}}{\sqrt{a-3}}\\ \)

So a has to be a perfect square and a-3 has to be a perfect square.

The smallest value of a to meet this criterion is a=4

I have just seen your answer ElectricPavlov.  YES your answer is correct :))