CPhill

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!!

Thumbs up, GoldenLeaf!!

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

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

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......

So we have

595(1 + .055/4)^4x4

Where

595 is the amt invested

.055 is the interest rate expressed as a decimal

The "4" inside the parenthesis represents the number of compoundings per year - 4

And the first exponent of "4" represents the time, in years

And the second exponent represents the compoundings per year - 4

So we have, in dollars

$$\mathrm{\ }$$$${\mathtt{595}}{\mathtt{\,\times\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{0.055}}}{{\mathtt{4}}}}\right)}^{\left({\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{4}}\right)} = {\mathtt{740.305\: \!270\: \!404\: \!234\: \!410\: \!8}}$$

≈ $740.31