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On the calculator here, press Enter or = twice.

For example if you do 3 + 1/2 and press = you get 3.5, but if you press = again you get 7/2.

x^2 + x= x^3  ???

I think you might be getting your laws of exponents confused

Note that (x^2)* (x) = x^3

To see why the first thing isn't true (and it has to hold for all real numbers), let x =2

So

2^2 + 2 = 2^3  ???

4 + 2 = 8   ???

No.....that definitely doesn't look right, does it???

So, in general.....aⁿ + a^m ≠ a^m+n  

0.75^(x+1) = 20

Lei's take the log of both sides...so we have

log .75^(x+1) = log 20     and by a "log" property, we can write

(x + 1) log . 75 = log 20   divide both sides by log .75

x + 1 =  log20 / log .75    

x + 1 = -10.413               subtract 1 from both sides

x = -11.413

Well, Jedithious....this is kind of a tough one, huh??

Let's call the amount of water contained in D originally "x"....since we don't know how much it contains.

After the first pour, D loses(1/2)x and C gets (1/2)x

x -(1/2)x = (1/2)x

So D now contains (1/2)x

And, on the next pour, D loses (1/2)(1/2) x  = (1/4)x and C gets (1/4)x

So D now contains x - (1/2)x - (1/4)x = (1/4)x

And C contains  (1/2)x + (1/4)x = (3/4)x

On the last pour D loses(1/2)(1/4)x = (1/8)x  and C gets (1/8)x

So....let's look at what C contains, now.

It has (1/2)x + (1/4)x + (1/ 8)x = (7/ 8)x

But.....(7/8)x = (1/2)C = (1/2)D ....since both glasses are the same size and shape.

Therefore

(7/8)x = (1/2)D  and since we want to find out what x was, let's multiply both sides by (8/7)

So we have

x = (8/7)(1/2)D = (8/14) = (4/7)D

So D was (4/7) full when we started

So, since each pour "halves" what comes before, we have (4/7)(1/2)(1/2)(1/2) = 4/56 = 1/14 full...and that's the fraction of the glassful remaining in D.

Whew!!...some workout, huh?  Let me know if you spot any mistakes......but I think that's it......a pretty challenging problem!!

Glass                  "C"                       "D"                Water Left

Initial                 0                         100

1st                     50                        50                    1/2

2nd              50+25=75                  25                     1/4

3rd              75+12.5=87.5            12.5                   1/8

4th               87.5+6.25=93.75       6.25                  1/16

"Two glasses, C and D, have exactly the same size and shape. Glass C is empty, and glass D has some water in it. Half the water in D is then poured in to C. This process is repeated two more times. Each time, half the remaining water in D is poured into C. After three pourings, C is half full. What fraction of a glassfull is now left in D?"

Suppose glass D starts out with d amount of water in it.

after 1 pouring glass C has d/2, and glass D has d/2 water left in whatever units

after the 2nd pouring C has (d/2+d/4) and D has d/4

after the 3rd pouring C has (d/2+d/4+d/8) and D has d/8

We are told that glass C now is half full.  

So 7/8d=1/2 a glass

d = 4/7 of a glass

and there is d/8 left in glass D at this point so

glass D now has 1/8(4/7) = 4/56 = 1/14 of a glassful

ohhh when you worked it out I understood it.. Thank you for helping me! 

Thumbs up, GoldenLeaf!!

The "excited" button.....much more apt description than "factorial," don't you think??

7*a - 5*b

7*7 - 5*2

49 - 10

39

$$\left({\frac{{\mathtt{2}}}{{\mathtt{x}}}}\right){\mathtt{\,\small\textbf+\,}}\left({\frac{{\mathtt{3}}}{{\mathtt{x}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}\right) = {\mathtt{0}}$$

$$\left({\mathtt{2}}\right){\mathtt{\,\small\textbf+\,}}\left({\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}}\right) = {\mathtt{0}}$$

$${\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{0}}$$

$${\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}} = {\mathtt{0}}$$

$${\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{x}} = -{\mathtt{5}}$$

$${\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{5}}}{{\mathtt{4}}}}$$

It's the excited button (as I call it). It really is the factorial, or, on the calculator here, the '!'

Writing 223%  as a decimal, we get

2.23 x 86.5 =

192.895

And that's it !!

8.5 =

(8 + 1/2) =

17/2

40 =

8 x 5 =

2 x 4 x 5 =

2 x 2 x 2 x 5 =

2³ x 5

And that's the prime factoring of 40

36% = 36/100 = 18/50 = 9/25

80 / 2 = 40, 140 - 100 = 40, 180 - 140 = 40, 560 - 520 =40

by the way sorry guys that i only answered a few questions today i can only be on this computer for a short period of time due to (outer inteferences)well and Melony and Cphill (mostly chris) question hogging lol jk but seriously they do alot show them so love maybe private messages thumbs up there the best online people i know lol but thats ok ill try to do better tommorow.....until then

See Yah guys zegroes

8 times 5 is one 40 times 1 is one 4 times 10 and 2 times 20 is one and you find this  by simply finding what times what equals 40  and this applies to all numbers by the way when trying to find factorizations

hey Cphill maybe the extra zero was left off due to rounding maybe?.....  (0.017)

that depends but the real question is does a chicken give birth to an egg or does an egg give birth to a chicken.....

1 - (-4) =   1 + 4 = 5

A negative outside any parentheses or brackets changes the sign of everything inside.

.051 / 3   = .017

So, the student is not correct......

 i think its infinite really..... 3.14 is just the round up