CPhill

41, 42, 45, 50, 57, 66, 77, 82, 83, 85, 88, 90, 92, 97   give non-primes  = 14

So

100 - 14  =  86

First one....we have that

n^2 + ( n + 1)^2  =  41

n^2 + n^2 + 2n + 1 = 41

2n^2 + 2n - 40  = 0

n^2 + n  - 20 =  0

(n  + 5) ( n - 4) = 0

Second one

Call the first factor, F

So...the second is   F/2 - 11

So.....this implies that

(F)  [ (F/2) - 11]   = -36       simplify

F^2/2 - 11F = - 36

F^2/2 - 11F + 36 = 0         multiply through by 2

F^2 - 22F + 72 = 0

(F - 18) ( F - 4)  = 0

F = 18      or F = 4

When F = 18    the other factor is :  F/2 - 11   =  18/2 - 11  =  9 - 11 =   -2

When F = 4    the other factor is   4/2 - 11 =  2 - 11  =  -9

So....the factors are

(18, -2)   and ( 4, - 9)

0 ≤  t  ≤ 10   ⇒  the domain

The divisors of 2008  are

1, 2, 4, 8, 251, 502, 1004, 2008

"b"  will be smallest  when   we have  251 and 8

So

(x + 251) ( x + 8)  =   x^2 + 259x + 2008

So....the smallest value of "b"  =   259

Let N be what he started with

After he gave1/3 of his coins to Clark  and 1/3 of the remaining coins to Lois  we have

After giving (1/3) to Clark, he had (2/3)N left

After giving (1/3) of these to Lois, we have

(2/3)[(2/3)N] = 144 

(4/9) N  = 144

N =  144 * 9 / 4   =  324

After giving (1/3) of these to Clark he had   216 left

He gave (1/3) of these to Lois =  216 / 3  =   72

OK....I see what you mean.....

The domain   is correct...( -inf, inf)

In the form

y = trig function (Bx + C)

Factor out B

y  = trig function B( x + C/B )

Set   x + C/B = 0   and solve for   X

Example

sin ( 3x - pi )

sin 3(x - pi/3)

Set

x + pi/3  = 0

x =  -pi/3          this is the phase shift

No graph is shown.....

Let the long side of a rectangle = x

Let the short side   =  y

By the diagram....it must be that    3x = 4y   ⇒  x = (4/3)y

So  L =  3x  = 4y

And W = 2y + x   =  2y + (4/3)y  =  (2 + 4/3)y = (10/3)y

So

L / W   =    (4y) /  [ (10/3) y ]  =    4 / (10/3)  =   12 / 10  =   6 / 5

2^(2/3) =  (2^2)^(1/3)  = (4)^(1/3) 

2^(-5/4)

Note that this is  (2/1)^(-5/4)

We can "flip" the fraction and make the exponent positive.....so we have

(1/2)^(5/4)  =   [(1/2)^5 ] ^(1/4)   =  [ 1/32 ]^(1/4)   

The last answer might also be simplified as    [ (1/16) (1/2) ]  ^(1/4)  =  

(1/16)^(1/4) * (1/2)^(1/4) =

(1/2)* (1/2)^(1/4)