MaxWong

log 10 + log 10 + log 10 + log 10 = 4, and 10 - log 10 - log 10 - log 10 = 7 perhaps?

:P

log is a function, not a number.

Still 0.

In this case, you think of a number that when it multiplies itself, it gives out 0.

0 * 0 = 0, so \(\sqrt0\) = 0.

\(11011_b = (b^4 + b^3 + b + 1)_{10}\\ \quad 11011_b \times (b-1)\\ =\left((b^4+b^3+b+1)(b-1)\right)_{10}\\ =(b^5 - b^3 + b^2 - 1)_{10}\\ \\ (b^5-b^3+b^2-1) + (b^3+1) = b^5 + b^2\\ (b^5+b^2)_{10} = 100100_b\)

Done! Yay!

f(x) = ax + b

x = a*f^-1(x) + b

f^-1(x) = (1/a) * x - b/a

so you get 1/a = b, -b/a = a

because -b/a = a, b = -a^2

Only solution is a = -1, b = -1.

so a + b = -2.

ab + a + b = 524

ab + a + b + 1  = 525

(a+1)(b+1) = 525.

bc + b + c = 146

(b+1)(c+1) = 147.

cd + c +d = 104

(c+1)(d+1) = 105.

525 = 3 * 5^2 * 7

147 = 3 * 7^2

105 = 3*5*7

Most likely b+1 = 21, c+1 = 7 by finding GCD.

so b = 20, c = 6.

Now we find a and d using the values of b and c and the equations bolded.

21(a+1) = 525

a+1 = 25

a = 24

7(d+1) = 105

d+1 = 15

d = 14

So, a-d = 24 - 14 = 10

Merry Christmas!!

-6x + 220 = -566

-6x = -786

x = 131

0.8m - 3.2 = 8 - 4.8m

5.6m = 11.2

m = 2

151u + 411 = 203 - 109u

260u = -208

u = -0.8

-4(b+2) + b = -4(1+b)

-4b - 8 + b = -4 - 4b

-8 + b = -4

b = 4

\(\because e^{i\pi} = -1\\ \text{We can immediately imply that} \log(-1) = i\pi\\ \dfrac{1}{6}(-i\log(-1))^2 \\ =\dfrac{1}{6}(-i\cdot i\pi)^2\quad\boxed{-i\cdot i = 1}\\ =\dfrac{1}{6}\pi^2\\ \)

Hence, proved.

a^5 + b^5 = (a+b)(a^4-a^3*b+a^2*b^2-a*b^3+b^4)

Easier divisibility rule for 11:

if absolute value of ((1st digit + 3rd digit + 5th digit + ....) - (2nd digit + 4th digit + 6th digit + ...)) = 0 or anything divisible by 11,

then the number is divisible by 11.

Example:

Is 43956 divisible by 11?

(4 + 9 + 6) - (3 + 5) = 19 - 8 = 11 <--- divisible by 11.

So 43956 is divisible by 11.

Is 10293583762 divisible by 11?

(1 + 2 + 3 + 8 + 7 + 2) - (0 + 9 + 5 + 3 + 6) = 23 - 23 = 0 

So 10293583762 is divisible by 11.

Did you mean:

Homeomorphic

Mobius strip?

I don't know anything about topology, but I think a torus is homeomorphic to the Mobius strip.

I just don't know... lol

It is not just divide by (power + 1) and add 1 to the power.

Examples:

\(\displaystyle\int\cos x dx = \sin x + C\\ \displaystyle\int \sec^2 x dx = \tan x + C\\ \text{OR even more complicated:}\\ \displaystyle\int \sqrt[3]{\tan x}dx = -\dfrac{1}{2}\ln|1+\tan^{2/3}x|+\dfrac{1}{4}|\tan^{4/3}x-\tan^{2/3}x + 1|+\dfrac{\sqrt{3}}{2}\arctan\left(\dfrac{2\tan^{2/3}x-1}{\sqrt3}\right) + C\)

\(7\cdot 3^{x+1} = 2+\dfrac{3}{3^{x}}\\ 21\cdot 3^{2x}-2\cdot 3^x - 3=0\\ \boxed{u=3^x}\\ 21u^2 - 2u -3 = 0\\ (3u+1)(7u-3)=0\\ u = \dfrac{-1}{3}(rej.) \quad OR\quad u = \dfrac{3}{7}\\ 3^{x-1} = \dfrac{1}{7}\\ (x-1) = \log_3\dfrac{1}{7}\\ x = 1 + \log_3(7^{-1})\\ x = 1 - \log_3(7)\)

f(0) = c so c = 4,

f(1) = a + b + 4, implies that a + b = 11, so 2a + 2b = 22 --- (1)

f(2) = 4a + 2b + 4, implies that 4a + 2b = 32 --- (2)

(2) - (1) gives 2a = 10, a = 5.

5 + b = 11, b = 6.

Therefore abc = 4*5*6 = 120.

Good morning to you, but good night to me.

Here is Hong Kong, and it is 22:28PM. 

My day has been good, favourite lessons all in a day... :D

Programming, Chemistry, Physics.... :D