CPhill

This works just like "long  division".......we have

                      x^2   - 4x  + 2  

         ----------------------------

x+5  ) x^3 +   x^2 - 18x +  2

          x^3  + 5x^2

         ----------------------------

                   -4x^2  - 18x

                   -4x^2  - 20x

                   ---------------

                                 2x  +  2

                                 2x  + 10

                                 -----------

                                          -8

So, there's our answer......  x^2 - 4x  + 2  with a "remainder" of -8 /(x+5)

Note that if we can factor the numerator/denominator, this sometimes "cancels" terms and division might not be necessary at all, but that wasn't possible here.

(1,-4) (5,2)

We first need to find the slope. This is just the change in the y coordinates divided by the change in the x coordinates. Note that y changes from -4 to 2 so that's 6. And x changes from 1 to 5 and that's 4.

So (change in y)/ (change in x)   =   6/4  =  3/2   and that's the slope.

To write the equation of the line we can use one of the points and the slope. Using the "point-slope" form, we have

y - y₁ = m(x -x₁)    where m is the slope and x₁ and y₁ are the coordinates of one of the points.

I'll use (5,2) since there is no negative involved (but I could have used either point)....so we have

y - 2 = (3/2)(x - 5)       simplify the right side

y - 2 = (3/2)x - 15/2       add 2 to both sides

y =  (3/2)x - 15/2 + 2   simplify the right side

y = (3/2)x - 11/2

And there's the equation of the line in "slope-intercept" form.

I like this form because the slope tells me if the line 'falls or rises" from "left to right" and I can also see the y intercept (0, 11/2) right away.

-10x-8y=7

30x+24y=-20

We can quickly see that this sytem has no solution....if we multiply the first equation by -3 on both sides, we get:

30x  + 24y = -21

This says that the left side = -21

But the second equation says that the same left side = -20.

And this is impossible. Adding two terms together can't produce two different answers.

-2(3x +5) = -3(4x + 3)

Simplifying on the right and the left, we have

-6x -10 = -12x - 9

Add 12x to both sides

6x - 10 = -9    

Add 10 to both sides

6x = 1

Divide both sides by 6

x = 1/6

Let's check this in the original problem to make sure it "works"

-2[3(1/6) +5] = -3[4(1/6) + 3]    ???

-2[1/2 + 5] = -3[2/3 + 3]  ???

-2[11/2] = -3[11/3]   ???

- 11  =  -11   ....and that's true !!!

Let's examine how we can arrive at  Alan's  answer,  since it might not be immediately intuitive........

Note that  log 3⁵⁰⁰   asks.........What power do we raise 10 to in order to obtain 3⁵⁰⁰  ???

By a property of logs, we can write

log 3⁵⁰⁰    as   500*log3 .........and evaluating this gives us......... 238.56062735983122

Now....note that the "whole number" part of this tells us what power (exponent) will be on the "10" in the scientific notation representation of 3⁵⁰⁰ , and 10 raised to the decimal part will tell us what some of the first few digits of that number will be (within a certain degree of accuracy).

So 10^{.56062735983122} = 3.63602917958700501  (Only the first 12 digits of this are accurate for 3⁵⁰⁰ ... see below)

Thus, in scientific notation we would have......

3.63602917958 X 10²³⁸   

This number is actually 239 digits in length and Alan's answer of   3.636×10²³⁸   is good enough !!!!

However....if you insist.....we have

3636029179586993684238526707954331911802338502600162304034603583258060019158389548419850

8262979388783308179702534403855752855931517013066142992430916562025780021771247847643450

125342836565813209972590371590152578728008385990139795377610001

 I think I'll stick with the scientific notation........

You're dull and uninteresting, biderman......nobody is paying any attention to your (supposedly) humorous posts....disappear back to troll land.......where your posts will actually display.......

We have :

[3.09 x 10^(-9)] / [.-02 x 10^(-9)]

Notice that the 10^(-9) terms "cancel" and we're left with

3.09/.02     

If we multiplied the top and bottom by 100, the decimal points in the numerator and denominator would move to the right by 2 places, and we get

309/2

And there you go.....

Just enter this into the on-site calculator and hit " = "

Thanks for those resources, Melody.......I find that people who don't know the "times table" to some reasonable extent - whatever one defines that as - have trouble reducing fractions, simplifying radicals......etc.

I just did something like it......look under "Answers"