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don't cheat on aops problems guys

2. Factor 4x^2y^2-(x^2+y^2-z^2)^2 as the product of four polynomials of degree 1. 

4x^2y^2-(x^2+y^2-z^2)^2  =

[2xy + ( x^2 + y^2 - z^2) ]  [  2xy - (x^2 + y^2 - z^2)]  =

[[ x^2 + 2xy + y^2] - z^2 ] [ 2xy - x^2 -y^2 + z^2 ]  =

[ (x + y)^2 - z^2]  (-1)[ x^2 - 2xy + y^2 - z^2]  =

[ (x + y)^2 - z^2] (-1) [ ( x - y)^2 - z^2]  =

[ (x + y)^2 - z^2] [ z^2 - (x - y)^2]  =

[(x + y) + z ] [ (x + y - z] [ z + (x - y)] [ z - (x - y)]  =

[ x + y + z ] [ x + y - z ] [ x - y + z] [ -x + y + z ]

1. Factor p^2-2p+1-q^2-2qr-r^2 as the product of two polynomials of degree 1.

p^2-2p+1-q^2-2qr-r^2 a  =

(p - 1)^2 - (q  + r)^2  =

[ (p -1)  + (q + r) ] [ ( p - 1) - (q + r)]  =

[ p + q + r - 1]  [ p - q - r - 1 ]

7,999,999,999 =7999999999 = 1999 * 4002001

2ab-14a+5b-35  =

(2a + 5) (b - 7)

xy+3x-4y-12  =

(x - 4)(y  + 3)

1,025,110P2=1,025,110 x 1,025,109 =1,050,849,486,990 - 2-word permutations.

That is the calculator method.

How to work out 8^(5/3) manually.

\(8^{\frac{5}{3}}=(\sqrt[3]{8})^5=2^5=2\cdot 2 \cdot 2 \cdot 2 \cdot 2 = 4 \cdot 2 \cdot 2 \cdot 2 = 8 \cdot 2 \cdot 2 = 16 \cdot 2 = 32\\ \text{Or if you know what's 2^5, the answer is straight-foward-ly 32.}\)

See your latest post. Melody replied to your post about the derivative one.

Good job Guest #2

\(\dfrac{d \;f(x)}{dx}=1x^{-1}\\ \dfrac{d}{dx} f(x)=\dfrac{1}{x}\\ f(x) = \int \dfrac{dx}{x}\\\quad\;\;\;\;\!=\ln|x|\)

\(\dfrac{d}{dx} 9\ln|x|=\dfrac{9}{x}\)

Melody's way is a good way but I never knew this. Thanks Melody for presenting this method.

\(I=\int x \cos(x) dx\\ \begin{array}{rl}u=\cos x & du=-\sin x dx\\ dv=xdx&v=\dfrac{x^2}{2}\end{array}\)

\(\text{Then }\)

\(\begin{array}{rll}I=\int udv & u\cdot v = \dfrac{x^2\cos x}{2}&\int vdu = \int\dfrac{-x^2\sin x}{2}dx\\\end{array}\)

And I am stuck??

What if I am stuck in this situation? Do I do integration by parts again?

1,025,109.8 words?
Do you mean 10,251,098 words? Or that would be nonsense

If the derivative of a function f(x) is 1x^(-1), what is the derivative of 9f(x)?

\(If\\ \frac{d}{dx}\;f(x)=\frac{1}{x}\\ then\\ \frac{d}{dx}\;[9f(x)]=\frac{9}{x}\\\)

If the derivative of a function f(x) is 1x^(-1), what is the derivative of 9f(x)?

EVALUATE THE EXPRESSION:
\((2+1)(2^2+1)(2^4+1)...(2^{1024}+1)+1\)

\(\begin{array}{|rcll|} \hline (2^{\color{red}1}+1)(2^{\color{red}2}+1) &=& 2^{\color{red}3}+2^2+2^1+1 &| 1+2 = 3\\ (2^{\color{red}1}+1)(2^{\color{red}2}+1)(2^{\color{red}4}+1)&=& 2^{\color{red}7}+2^6+2^5+2^4+2^3+2^2+2^1+1 &| 1+2+4=7\\ \dots \\ (2^1+1)(2^2+1)(2^4+1)...(2^{1024}+1) &=& 2^{2047}+2^{2046}+\dots +2^1 + 1 \\ &&|1+2+4+\dots + 1024 = 2047 \\\\ & & 2^{2047}+2^{2046}+\dots +2^1 + 1 = 2^{2048}-1 \\\\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline \mathbf{ (2^1+1)(2^2+1)(2^4+1)...(2^{1024}+1)+1 } & \mathbf{=} & \mathbf{2^{2048}}\\ \hline \end{array}\)

8^(5/3) =8^5 =32,768^(1/3)=32

i^2016 + i^2015 + i^2014........i^2 + i =0

Length of 1 degree of Longitude = cosine (latitude) * length of degree (Kilometers) at equator ,

111.1 x cos-1(1.55720185)

Use cos and NOT cos^-1

111.1 x cos(1.55720185) =111.1 x 0.999630=111.06 Km at equator.

 The title was supposed to be Imaginary Numbers 2...

how come when doing 100sin120/sin90which should equal 87 its giving me the wrong answer, how do I correctly enter this into the calculator to get the correct answer

\(\frac{100sin120}{sin90}\\ =\frac{100sin60}{1}\\ =100*\frac{\sqrt3}{2}\\ =50\sqrt3\\ \approx 86.6 \)

((100*sin(120))/sin(90)) = 86.6025403784

Perhaps you do not have your calculator set to degrees  ???

What is 2(a+8) expanded and simplified?

= 2 * a      +       2 * 8

= 2a +16

4) 1^81.2^91.6^27 =2.5340394e+48 mod 7 =5

1.2 x 10^7μm

wat does this equal

Find the minimum and maximum possible areas of a rectangle measuring 2 km by 5 km.

I think that this question relates to significant figures.  I will assume that each of these is rounded to the nearest km

so

minimum area

1.5*4.5 = 6.75 km^2

maximum area

2.5*5.5 = 13.75 km^2