All Answers 358087 Answers

     1                       1                 1                                      1            

________    +    ______  +   _______    +   ...... .+      _________    

√2 + √1             √3 +√2           √4 + √3                         √25 + √24

Multiply  numerator/denominator of each fraction by the  conjugate of its denominator and we get that

√2  -√1            √3 - √2        √4  - √3                            √25  - √24

______   +     ______   +   _______   +   ......  +      __________    =

   1                      1                   1                                      1

- √1  +  ( √2 - √2)  +  ( √3 - √3)   + ( √4  - √4)  +  ......+   (√24  - √24)   + √25    =

√25   - √1    =

5   -  1    =

4

sin (40°)  + cos(40°)  = sqrt ( 1  + sin x)            square both sides

sin^2(40°)  + 2sin(40°)cos(40°)   + cos^2 (40°)   =  1  + sin x

sin^2 (40°) + cos^2((40°) +  2sin(40°)cos(40°)  = 1   + sin x

(1)   + 2sin(40°)cos(40°)  =   1  +   sin  x

2sin(40°)cos(40°) =  sin x

sin ( 80°)  =  sin x

x = 80°

The ratio of the shaded triangle to the square is  1 : 10

( √x - 1) * √x * *( √x + 1)  = 6

Let  √x =  a  .....so we have

(a - 1)  * a  * ( a + 1)   =  6

(a - 1) (a + 1) a  =  6

(a^2 - 1) a  = 6

a^3  - a   = 6

a^3  - a  - 6   =  0         

By the rational roots theorem, the possible  values for a  = ± [ 1, 2, 3, 6 ]

By inspection     a =   2

Therefore   √x  =  2   ⇒    x  = 4

The area of the triangle on the left is 7.794 units², and the area of the triangle on the right is  10.394 units².

Therefore, the total shaded area is:  18.188 units²

We have  4 hairstyles  ( 3 plus bald)

2 sets of eyebrows

1 set of eyes

2sets of ears

2sets of lips

2 types of shoes

So....the possible  wacky personalities =   4 x 2 x 1 x 2 x 2 x 2  =  64

sum_(n=1)^24 1/(sqrt(n) + sqrt(1 + n))≈4

I'm stuck trying to understand this. 

1.  How do you tell that the shaded triangles are right triangles?

2.  It looks like the 6 that you're multiplying by is the hypotenuse of those shaded triangles instead of the height.

I'm sorry for being so dense.

_.

You mix apples and oranges.

You cannot subtract area from a line!

9! - [5! * 4!] =360,000 ways.

Do jou need the red area?

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91 =18 such numbers.

a=1;b=1;c=2;p=0; cycle:d=a*100+b*10+c;if(d%2==0 and d<500, goto loop, goto next); loop:printd," ",;p=p+1; next:c++;if(c<6, goto cycle, 0);c=2;b++;if(b<6, goto cycle, 0);b=1;c=2;a++;if(a<10, goto cycle,0);print"Total = ",p

OUTPUT: 112  114  122  124  132  134  142  144  152  154  212  214  222  224  232  234  242  244  252  254  312  314  322  324  332  334  342  344  352  354  412  414  422  424  432  434  442  444  452  454  Total =  40 such numbers.

Here are the possibilities

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111   16 possibilities    6 have 2 of each     6/16 = 37.5 %    ...as Guest posted!

The probability is: [4! /2!. 2!] =6 / 2^4 =6/16 =37.50 %

If your question means "any one element that is divisible by 2, 3, 5, 7 at the same time, or divisible by all four", then there is only one such element:

210 mod [2, 3, 5, 7] = 0

If your question means "any one element that is divisible by ANY ONE of the four (2, 3, 5, 7), then All of them except 1 are divisible by at least one of the four, or 279 numbers.

I've only solved part (a)... . The address for the mathisfun solution is https://www.mathisfunforum.com/viewtopic.php?id=22435 

First, we choose the two women to stand next to each other.  We can do this in 5*4/2 = 10 ways.  There are then 8 ways to choose the two places where they stand next to each other.  There are 7 other people on the committee, so there are 10*8*7! ways to arrange the members.  Answer = 403200.

Since g(z) is equidistance from 0 and z, |g(z)| = |g(z) + z|. This leads to a = -4.

Since z = 2e^(i*pi//6) and w = 5e^(i*2*pi/3), z/w = 2/5e^(-5*pi/6).  Therefore, r = 2/5.

There are total 216 outcomes if we could calculate it manually, It is better to use online probability calculators like these:

Solve (sqrt(x) - 1)*sqrt(x)*(sqrt(x) + 1) = 6.

(sqrtx-1)*(x+sqrtx)

x sqrtx +x -x-sqrtx = 6

x sqrtx - sqrtx = 6

x^3/2 - x^1/2 = 6    let x = x^1/2  

x^3-x = 6

x(x^2-1) = 6

x(x-1)(x+1) = 6                               Maybe this: assuming x is positive,    factors of 6 are 1 2 3 6  

                                                           then you can deduce x = 2

                                                                 then since we substituted x = x^1/2   ,    x^1/2 =2     or   x = 4

x^1/2 (x^1/2-1)(x^1/2+1) = 6

    x = 4    (from wolframalpha....I did not know how to proceed)

(sqrt(x) - 1)*sqrt(x)*(sqrt(x) + 1)   = 6

                                               x   = 4

(sqrt(4) - 1)*sqrt(4)*(sqrt(4) + 1) = 6

                          (2 - 1)*2*(2 + 1) = 6

                                          1*2*3 = 6