CPhill

Solution set

Nice, guest   !!!!

We have a couple of things to consider

First.....the domain of the function ih the numerator is  x ≥ 0

For the  function in the denominator.....1 - x^2  must be greater than 0

So

1 - x^2 > 0

( 1 - x) ( 1 + x ) >  0

If     -1 < x < 1     this is true

But....the restriction in the denominator makes the true domain   0  ≤ x < 1

√ [3x] - 4√3               2√[2x ]  + √2

__________   =       ____________

√x  - √2                      √[6x] - 2√3

√ [3x] - √48              √[8x ]  + √2

__________   =       ____________     cross multiply

√x  - √2                      √[6x] - √12

( √ [3x] - √48 ) (  √[6x] - √12) =  (√x  - √2) ( √[8x ]  + √2)     simplify

3x√2  - 12√[2x] - 6√x + 24   =  2x√2 -  4√x + √[2x] - 2

x√2 - 13√[2x]  - 2√x + 26  = 0

√2 √x √x -  13√2 √x  - 2√x + 26 = 0      factor as

√2√x [ √x  - 13 ] - 2 [ √x - 13]  = 0

[ √x - 13]  [ √[2x] - 2 ]  = 0

So either

√x - 13  = 0                                   or         √[2x] - 2  = 0

√x = 13   square both sides                       √[2x]  = 2        square both sides

x = 169                                                          2x   = 4

                                                                       x = 2

Reject the second solution....it makes the original denominators = 0

So

x = 169

We have a right triangle with   a hypotenuse of   6371 + 6.6  =  6377.6 km

The Earth's radius is one of the legs = 6371 km

And the distance to the horizon is the other leg and is given by :

√ [ 6377.6^2 -6371^2 ]  ≈   290.1 km

³√x^8                       x^(8/3)

__________   =    ________    =  x^(8/3) - x^(2/3) )   =  x^(6/3)   = x^2

 ( X^2)^(1/3)           x^(2/3)

"a" can be any integer  because 2a  is always divisible by 2....the requirement for "eveness"

(f - y)  = [ 2x] - [ 3 - x ] =   3x - 3

(f - y) (4)  =   3(4) - 3   =  12 - 3   = 9

y = x + 5

y = -2x -1

Set the y's equal

x + 5 =  -2x - 1      add2x to both sides, subtract 5 from both sides

3x = - 6     divide both sides by   3  

x = -2

And using  y = x + 5 to find y, we have

y = (-2) + 5

y = 3

Newton's first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.