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cube 1  3 x 3 x 3      

cube 2  4 x 4 x 4

One 3x3 face will be gone (covered)  as will a  3x3 area on one of the faces of the larger cube

3x3   x 5        +   4x4 x 6  - 3x3 = 45 +87=132 cm²

Original painted surface area =  2 ( 3 x 6  + 6x8 + 8x3)   180 cm^2

After making it 1 cm^3   there are   3 x 6 x 8 cubes   each with 6 cm^2 surface area = 864 cm^2

864-180 = 684 cm^2 unpainted

In steps

Note that (x + 2)^2  (x - 2)^2   can be written as

[  (x + 2) ( x - 2) ] ^2   =   ( x^2 - 4]^2  =  x^4 - 8x^2 + 16

So

(x - 3)   (x^4 - 8x^2   + 16 )  =

(x^5 -8x^3 + 16x) + ( -3x^4 + 24x^2 - 48)  =

1x^5 - 3x^4  - 8x^3  + 24x^2 + 16x  - 48

Now  we just need to multiply

(1) (-3) (-8) (24) (16) (-48)   =  -442368

1 = LCM of 90 and 8 = 360
GCD of 90 and 8 = 2
2 = LCM of 72 and 10 = 360
GCD of 72 and 10 = 2
3 = LCM of 40 and 18 = 360
GCD of 40 and 18 = 2
4 = LCM of 18 and 40 = 360
GCD of 18 and 40 = 2
5 = LCM of 10 and 72 = 360
GCD of 10 and 72 = 2
6 = LCM of 8 and 90 = 360
GCD of 8 and 90 = 2

9.95p + 8.01p <= 50  , so p <= 2      This answer is not listed....did you make a typing mistake for 'A' ?

Area =  pi * r^2 *  (30/360)  =

pi * 8^2  *  1/12  =

64/12  * pi  =

16/3 pi ≈  16.76  ( units squared )

Note  that  1 * 2 * 3 * 4 * 5  =  5!  =   120 = (60 * 2)

Then  6!  =  5!  * 6  =   (60 * 2) * 6 =  60 (12)

And  every factorial from 7 to 100  will contain  5!  =   60 * 2

So from   1!  to  100!  we  will have

100 - 5  + 1   =     96 factorials that are divisible by 60

Note that   (1 + sqrt2 )     and ( 3 -sqrt 5)  are also  roots

We have

(x - (1-sqrt 2) )  ( x - ( 1 + sqrt 2) )   ( x - (3 + sqrt 5) )  ( x - ( 3 + sqrt 5)

Simplify in steps

( x - ( 1 -sqrt2) )  ( x - (1 + sqrt 2))  =  

x^2 - [ ( 1 -sqrt 2) + ( (1 + sqrt 2)] x  + ( 1 -sqrt 2) (1 + sqrt 2) =

x^2 - [ 2 ] x  + 1  - 2  =

x^2 - 2x - 1

Similarly

( x - ( 3 + sqrt 5) )  ( x - (3 - sqrt 5))  =

x^2  - [( 3-sqrt 5) + (3 + sqrt 5)]x   +  (3 +sqrt 5) (3 - sqrt 5)  =

x^2 - 6x + 9 - 5  =

x^2 - 6x + 4

So

(x^2 - 2x -1) ( x^2 - 6x + 4)  =

    ( x^4 -2x^3 - x^2 )

 +        ( -6x^3 + 12x^2 + 6x)

 +                    (    4x^2  -8x  - 4)   =

x^4  - 8x^3 + 15x^2  -2x   - 4

OUTPUT =  3n + 1

exactly.

TWO digit numbers that are 1 greater than a multiple of 8 would be  17 25 .....   as guest posted

Adding 6 to x shifts things to the LEFT:

y = 3 (x+6)^2 + 5(x+6) + 2

y = 3x^2 +36x+ 108  + 5x + 30 + 2

y = 3x^2 + 41x + 140

Note that triangles  BAC  and BFD  are similar

Since AB = 12, let the side of the square  = x =  AF

Then BF  =  12  - x      ....and  by similar  triangles we  have that

AB / FB  =  AC / DF

12 / ( 12- x)  =  18 / x            .....cross-multiply

12x  =  ( 12 - x ) 18

12x = 216 - 18x        ......  add 18x to both sides

30x =  216               .......divide both sides by 30

x = 216 / 30   =  7.2 =  AF

17 , 25 , 33 , 41 , 49 , 57 , 65 , 73 , 81 , 89 , 97 , Total =  11 such integers.

thank you!

Wendy and 4 friends  = 5 people

2 apples  / 5 people  =    2/5   =  portion of an  apple that each received

The remainder is 2.

You are given the 'zeroes'  (x axis intercepts),  ( -3 and 8)  so the equation is this :

y = ( x+3)(x-8)    <=====  expand to the form requested

2.5 ton * 2000 lb/ton  / (20 lb/lot) =     2.5 * 2000 /20    lots = _______________lots

The vertical axis of symmetry is  x = -12    the x intercepts are on either side of this at the same distance

   the second intercept will be at x = 6                      (   6,0   )

Oh no is he dead? I thought you was dead!

 I SHOULD be out 'there' putting my car back together that is scattered all over the driveway.....but it is still kind of cold....maybe later...haha...

Output = n + 4

Thanks, EP.....I've been away a LONG time.....a lot of life changes have happened.....great to be back and see that you're still "here"  !!!!!

Here's an example  it's called the "divide and average"  method

Sqrt   13       (to three decimal places)  = 3.605

First since  the sqrt  9   is 3    and the square root of 16 is 4......let's guess that the sqrt 13  might be something like 3.5 ( actually......we can make any "educated guess"....it will still work out

1)  Divide 13  by  3.5   =  3.174

2)  Take the average  of  3.5  and 3174 = 3.337

3) Divide 13 by 3.337  = 3.895

4) Take the average of  3.337 and 3.895 = 3.632

5) Divide 13  by 3.632  = 3.579

6) Average 3.632 and 3.579  = 3.605

And our job is complete  !!!!........hope that might help you  !!!