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\( sqrt(3 - sqrt(8 - sqrt(x)))\)

The least   value in  the domain  will  be  when x = 0

So we have  

sqrt ( 3 - sqrt ( 8 - sqrt ( 0 ) ) )  =  sqrt ( 3 - sqrt 8 ) =  sqrt ( 3 - 2sqrt 2)

Since we cannot take the square root of a negative, the greatest value in the domain = 64  because

sqrt ( 3 - sqrt ( 8 - sqrt (64)) )  =  sqrt ( 3 - sqrt ( 8  - 8) ) = sqrt ( 3 - 0 ) =  sqrt 3

See the graph here : https://www.desmos.com/calculator/0nd567etzs

TYSM!

Nice job, BB   !!!!

\((\sqrt{600} + \sqrt{150} + 4\sqrt{24})/(6\sqrt{32} - 3\sqrt{50} - \sqrt{2})*8 = a\sqrt{b}\)

Let's take this  one term at a  time

sqrt (600) =  sqrt (100 * 6)  = 10sqrt 6

sqrt (150 )  =    sqrt ( 25 *6) = 5sqrt 6

4sqrt (24)  =  4 sqrt ( 4 * 6)  = 4*2 sqrt 6  = 8 sqrt 6

Adding these three terms we get   23 sqrt 6

Multiplying by 8  we get   184 sqrt 6

Now....look at the terms in the  denominator

6sqrt ( 32)  =  6sqrt ( 16 *2)  = 6*4 sqrt ( 2) =  24sqrt 2

3sqrt (50)  =  3 sqrt (25 * 2) = 3*5 sqrt (2)  = 15sqrt 2

sqrt ( 2) = 1 * sqrt (2)

So the denominator =  (24 - 15 - 1) sqrt 2   =  8sqrt 2

So  we have    184 sqrt 6  / ( 8 sqrt 2)  =  (184/8) sqrt 6 /sqrt 2 =  (23)sqrt 3

a + b =   23  +  3 =  26

The time  for the cyclist  to  ride the first segment  is    30 km / (9km / hr) = 30/9 hr = 10/3 hr

Call the  rate that the cyclist must average in the second segment = R

The total time for the second segment =  20  km / R km / hr   =  (20/R) hr

The total distance traveled =   50 km

So

Total Distance  / ( Total Time  ) =   Avg  Rate

So

50  /(10/3  + 20/R)  =  10

50 /  10  =   (10/3 + 20/R)

5 = 10/3 + 20/R

5 - 10/3  = 20/R

5/3 = 20 / R

R = 20 / (5/3)  =  20 * 3/5  =    60 / 5   =   12 km/ hr 

I'll give this problem a shot...

There are 1000 x 999 x 998 x 997 options. 

There are 200 x 199 x 198 x 197 successes. 

So the answer is (200 x 199 x 198 x 197)/ (1000 x 999 x 998 x 997)

30 km / 9 km / hr =   10 / 3  hr

(30 km + 20 km )   = 50 km       50 km / 10km/hr = 5 hours

5 - 10/3 =   5/ 3 hr   to cover 20 km

20 km / 5/3 hr =   12 km / hr

4 ( x + 7)(2-x)  + 4(x + 7) (6-x)   =

4[ 2x + 14 -x^2 - 7x ]  + 4 [ 6x + 42 -x^2 -7x ]  =

4 [ -x^2 -5x + 14 ] +  4 [ -x^2 -x + 42 ]

-4x^2 -20x + 56  - 4x^2 -4x + 168

-8x^2 - 24x  + 224

This is a parabola with the x coordinate of the vertex given by  -b/ (2a)  =  24 / ( 2 *-8)  = 24/ -16  =- 1.5

And  y coordinate of the  vertex is the max =  -8(-1.5)^2  -24(-1.5) + 224 = -18 + 36 + 224  =    242

OK

Let C = (-14,8)   and A  =  (4,8)

The midpoint of this  segment is   [  ( 4+ - 14) / 2  ,  ( 8 + 8)/2 ]  =   (-5, 8)

And this  median will pass through B =   (2, -6)

The slope of this median line   =  (-6-8)/(2- - 5)  = -14/7 =- 2

Using the midpoint we can write the equations of the line as

(y - 8) = -2 ( x - -5)

y = -2x -10 + 8

y = -2x -2

Let the midpoint between A and B be [ (4+2 ) /2 ,   (8-6)/2  ] = ( 3, 1)   

And this median will pass through C = ( -14,8)

The slope of this median line is  ( 1-8) / ( 3 - -14)  = -  7/17

And using the midpoint the equation of this median line is

y -1 = - (7/17) ( x - 3)

y = - (7/17)x + 21/17  + 1

y = -(7/17)x + 21/17 + 17/17

y = - (7/17)x  + 38/17

Let the midpoint of  B and C  be  [ ( -14+2)/2 , (-6+8)/2 ]  = (-6, 1)

This median will pass through A = ( 4,8)

The slope of this median =   (1-8)  / ( -6-4)   =  -7 / -10   =  7/10

Using the median the equation of this median line is

y - 8 =  (7/10)(x  -4)

y = (7/10)x - 28/10  + 8

y = (7/10)x -28/10 + 80/10

y = (7/10)x + 52/10  =  .7x  + 5.2

If we set any two of the lines =, we can find the x coordinate of the intersection of the medians

-2x - 2  =  .7x  + 5.2

-2 -5.2  = 2x + .7x

-7.2 = 2.7x

x = -7.2 / 2.7   =  -72/27  =  -8/3

And the y coordinate of the intersection is

-2(-8/3) - 2  =  16/3  - 2  =  16/3 - 9/3 =  7/3

I would recommend you start by finding the angle of the n-sided polygon which is 360-90-120-60 = 90 degrees. Since a square is the only regular n-sided polygon with angles of 90 degrees, we know that n = 4.

...I don't get the point of posting this

Add up the exponents of x:

9/2  +  1/2 ==5

==x^5

4  x  1/2 ==2

So: [4x^(9/2)(1/2x^1/2)==2x^5

Happy Valentine's Day 

Quadratics aren't exactly my strong spot... but I would play around with the equation a bit. Factor the left side?

In x^2 + 12x + 4, what two numbers multiply to 4 and add up to 12?

a=1; b=1;d=GCD(a, 60) ;e=LCM(a, 60); if(d==b+5 and e==b*(b +5), goto5, goto6);printd,e, a, b; a++;if(a<500, goto2, 0);a=1;b++;if(b<500, goto2, discard=0; 

GCD of [a,  60] ==x + 5

LCM of [a, 60]==x * [x + 5]

a ==100

x == 15

GCD[100, 60] ==20 ==(x + 5)

LCM[100, 60]==300==x * (x + 5)

We want to solve this :

-4.9t^2  + 4t - .4 ≥ 8        subtract 8 from both  sides

-4.9t^2 + 4t  - 8.4 ≥  0     multiply through by -1

4.9t^2 - 4t  +  8.4  ≥ 0

Using x for t, look at the graph here :    https://www.desmos.com/calculator/g6yr1y8a7p

Notice that the height is ≥ 8m    between  ≈  .117 sec   and  ≈ .7  sec

+x³ - x³  =  0 x³

'3' is   5 /7 of the way from -2 to 5

5/7 th of the way from 7 to -8   is   a   =  -3 5/7

= 2t^2 -20t -36         Bowl shaped parabola (due to positive coefficient of x^2)   min occurs at t = - b/2a = - -20/(2*2) = 5

First die, any number==6/6

2nd die, probability that it is different fron the first==5/6

3rd die, probability that it is different from the first two ==4/6

4th die, probability that it is different from the first three ==3/6

Then, the overall probability that they are all different from each other==6/6 x 5/6 x 4/6 x 3/6==360 / 6^4

Therefore, the probability that at least 2 of them are the same==1 - [360 /1,296] ==13 /18 ==72.22%

The center of the circle is (3/2,5).

The center of the circle is (-15/2,7/3).

Since the LCM of {2, 5, 10, 12} ==60

Then there are: 10,000 / 60 + 1 ==167 integers that are divisible by all four divisors, 2, 5, 10, 12.