ElectricPavlov

2^10 * 4^20 * 8^40 =

2 ^10    (2^2)^20   (2^3)40

2^{(10 + 40 + 120)} = 2 ¹⁷⁰

eight.

y=-2x^2+18x-15.

y = 2 (x^2 + 9x)    - 15     'complete the squre' for x to get     y = a(x-h) + k       vertex form for a parabola

   = 2 (x+4.5)²  - 55.5               vertex is  ( - 4.5, -55.5 )     

= sqrt 2*2 * 3*5    * 2*2 * 3   * 7*3*3  * 7*3*2   x⁴

= x² sqrt  2^5 3^5 *  5 * 7^2

 = 252x²  sqrt( 30)

You need to 'complete the square for x '

2x^2 + 6x + 11

2 (x^2 + 3x)       + 11            take 1/2 of the 'x' coefficient :    1.5

2 ( x+1.5)²       + 11          if you expand this out , you will find that you have added   2 *  1.5²   to the expression....

                                                so you need to subtract that amount to keep the original equation

2(x+1.5)^2    -   2 * 1.5²      +11

2 ( x+1.5)²   + 6.5         is the final form.....  take it from here .....

When x = 1    y = 3

           x = 2   y = 5

            x = 3   y = 7                 y = 2x + 1

Twenty eight

5 1/2 = 11/2

 11/2 mod 1/3 = 0.1666666666666667    there will be 1/6 pound left over

11/2  / 1/3 =   11/2  *   3 = 33/2 =   16  1/2           sixteen bags    there willbe 1/2 of a 1/3 leftover = 1/6 lb

2x-5 ≤ 2x+6

-5≤6                              this is true for all 'x'      (-∞ , +∞)

Here is a picture....can you calculate the distance between the two labeled points?