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here. SORRY. no words to explain. how dumb. i am.

(3y - 5) / 6 =  (2y -1)*3 + 3        multiply through by 6

3y  - 5 = 18(2y - 1) + 18

3y - 5  = 36y - 18 + 18

3y - 5 =  36y

-5 =  36y - 3y

-5 =  33y

y =  -5 / 33

(n- 3)  +( n -2) + (n - 1)  + ( n ) +  (n+1) + ( n + 2) + (n + 3)  =   14 (n + 3)

7n = 14 (n + 3)     divide through by 7

n = 2 ( n + 3)

n = 2n + 6

n  -2n = 6

-n = 6

n = -6

The smallest integer  is   -6 - 3 =  -9

Set the functions  = 

x^2 + 4x  + 6 =  (1/2)x^2 + x + 4       simplify as

(1/2)x^2  + 3x  + 2   = 0      multiply through by 2

x^2 + 6x + 4  = 0     

x^2 + 6x    =   -4              complete the square on x

x^2 + 6x + 9 =  -4 +9

(x + 3)^2    =   5  take both  roots

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

x = sqrt (5)  - 3   =  c               x =   -sqrt (5) - 3   =  a

c   -   a  = 

2 sqrt (5)

3^{15} + 3^{11} + 3^7 + 3^3==14528268 = 22 × 33 × 17 × 41 × 193

The largest factor == 193

a) The answer is t^3 - 6t^2 + 11t - 6.

b) The only solutions (x,y,z) are (3,1,2) , (-4, 1, -3/2), and  (5/3, 9/4, 2).  You should be able to work it out from there.

1/3 * 120 = 40

1/4 * 120 = 30

1/6 * 120 = 20

1-(1/3 + 1/4 + 1/6) = 1 - (9/12)=1/4

1/4 * 120 = 30

2^2 mod 10 = 4

2^6 mod 10 = 4

2^10 mod 10 = 4

any 2^(4k+2) mod 10 = 4, k≥0, can you see why?

(118-2)/4 + 1 = 30

30 * 4 mod 10 = 120 mod 10 = 0 mod 10.

answer is 0

sumfor(n, 1, 30, 2^(4*n - 2))==354460 7988759775 6610768188 2741425220 mod 10 ==0 - the remainder.

Simplify as

sqrt  ( x^2  - 25)   +  sqrt ( 12 + x)

Domain of first  function is  ( inf ,-5]   U  [5 , inf)

Domain of second function  [-12, inf)

We have two domain  intervals   =  [ -12 , -5]    U  [5 , inf)

The width of the first  = 7

The width of the second is  infinite

Either

3x + 62  =  7x + 60

4x =  2 

x = 2/4 =  1/2

Or

7x + 60  =  5x + 50

7x - 5x = 50 -60     

2x  = -10    

x =  -5

Or

3x + 62  =  5x + 60

62 -60 = 5x  - 3x

2 = 2x

x = 1

The  perimeter  is minimized  when x  = - 5

48  + 25  +  25  =   

98 ft 

ab + c =   2   

bc + a =   2

ac + b =   2            

ab + c =  ac + b                       ab + c =   bc  +  a

ab - ac  =  b - c                         ab - bc = a - c

a ( b - c)  =  b - c                         b ( a - c)  = a - c

a  = ( b - c)  / ( b - c)                    b  = ( a - c)  / (a - c)

a =  1                                             b  = 1

So   c   also   =   1

So

a + b +  c   =   3

An isosceles right triangle is removed from each corner of a square piece of paper, as shown, to create a rectangle.  

"as shown" huh?  There's nothing shown. 

As long as this is a guessing game, my submission is either 36 or 144.  Pick one.    

First seven primes

2, 3 ,5 ,7 ,11, 13, 17

           2      3     5      7      11     13      17

2                 5     7      9      13     15      19

3         5               

5         7

7         9

11      13

13      15

17      19

There are 49  possible outcomes

14  are  odd

So

P (even sum)  =  ( 49 - 14 )  / 49  =    35 / 49  =   5  /  7

xy   = 24        (1)

xz =   48        (2)

yz  =  12        (3)

Divide  (2) by (1)      getting     z /  y  =  2     ⇒   z =   2y

Looking at (3)

y * 2y  =  12

2y^2   = 12

y^2  = 12 / 2   = 6

y = sqrt (6)

z = 2sqrt (6)  

Looking at (1)

xy =  24

x * sqrt (6)  =  24

x = 24  / sqrt (6)  =   4sqrt (6)

So

x + y +  z   =  

4sqrt (6) + sqrt (6)  + 2sqrt (6)  =

7 sqrt (6)

Triangle  AOB  is similar to triangle COD

Then   since  base  CD  =  38/25   of  base   AB,  then the  height  of  COD  = (38/25)  that of  triangle AOB

Then  there are  38 + 25  =    63 equal parts  to  the height of the trapezoid  ...... and 63/2  = 31.5  equal vertical parts on each side of the  midline

And  since the  height of AOB  contains 25  of  these equal parts......then  the height of triangle YOX  must = 31.5  - 25  = 6.5  of the equal parts

And triangles YOX  and COD  are similar

So

base  of  YOX  / base  of  COD     =     6.5  / 38   =    13/76

So  XY  = (13/76) CD  =    (13/76) ( 38) =  13 / 2    =  6.5  in

hey nerdie i know what you meant but you didn't say it right, 8² is 64 

log ₂ 3 * log ₃ 4

By the change of base theorem we can write

log 3        log 4              log 4

____ *   ______  =     ______  =      log ₂  4  =   2

log 2         log 3             log 2

Nothing shown......

nice

image not shown.

look below

What's your explanation?

The pattern is for each term to multiple by 5 to get the next term. 

Here's the sequence.

\(2, 10, 50, 250, 1250, 6250, 31250\)

Simple addition:

\(2+10+50+250+1250+6250+31250=39062\)

ANS \(\Rightarrow 39062\)

-5376 z^(9/2) - 1/z^(9/2) - 2016 z^(3/2) - 144/z^(3/2) - 2304 z^(15/2) + 512 z^9 + 4608 z^6 + 4032 z^3 + 18/z^3 + 672
The constant term is 672