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\(7c\sqrt{3c^2}\,=\,7c\sqrt3\sqrt{c^2} \,=\,7c\sqrt3\,*\,c \,=\,7c^2\sqrt3\)          

Hi Rarinstraw,

Not many people are comfortable answering physics questions BUT on top of that the writing in your pics is too small and too blury.  

I personally like to copy and paste the questions into my answers. I can't if it is a photo of print.

Don't wait up for me!! I never really learned how to do these "sum" things!!

I assume this is :

ln [ a * ∛b]  =

ln a  +  ln ∛b  =

ln a +   ln b^1/3  =

ln a  +  (1/3) ln b  =

  [3 ln a  +  ln b ] /  3

I don't know how to prove this.....but maybe hectictar does......if not.....hang on for heureka........he's usually pretty good at these kind of things  !!!!

[ I always learn something from his answers ]

g(x)  =  x² + 5x             Plug in  n  for  x .

g(n)  =  n² + 5n

    3[ g(n) ]                    Substitute  n² + 5n  in for  g(n)  .

=  3[ n² + 5n ]              Distribute the  3  to each term in parenthesees.

=  3n² + 15n

3[ g(n) ]  =  3n² + 15n

Thanks, hectictar......!!!!!!

This can also be looked  at  as a composition of functions

Where A  = x        B  = x +  1     and C  = x - 4

So   

( A ° B ° C) =

B(C)  =  (x - 4 ) + 1  =  x - 3

A ( B(C) ) =   A (x - 3)  =     [ x - 3 ]  is the conversion

1/3 + 2/3^2 + 3/3^3 + 4/3^4 + 5/3^5 +............+ k/3^k =3/4

OK, CPhill or hectictar: use your brilliant algebraic knowledge to prove this. I can't !!.

Hahaha!! It was easy once I tried adding them together!

Impressive, hectictar........!!!!!!

Even Columbo would have a hard time figuring this one out....!!!!!

\( $\frac{27}{125}, \frac{9}{25}, \frac{3}{5},\ldots$\)

The series is 

27          (1/3)^(n - 1)

____  *                             where n  is the nth term

125        (1/5)^(n - 1)

So.......the 6th  term  is

27 * (1/3)^5 /  [125 * (1/5)^5]  =

(27 / 243) / [ 125 /3125]  =

(1/9) / (1/25)  =

25/9

For example...a shirt that is size  3  in country  A  is size  3 + 1 = 4  in country  B .

A shirt that is size  4  in country  B  is size  4 - 4 = 0  in country  C .

So... a shirt that is size  3  in country  A  is size  0  in country  C .

A shirt that is size  x  in country  A  is size  (x + 1) - 4  in country  C .

f(x)  =  t( s(x) )  =  t( x + 1 )  =  (x + 1) - 4  =  x - 3

f(x)  =  x - 3

Yes absolutely, although there is a high probability that probablility questions isare more perilous than most other types of questions.   LOL

I think I got it!!!!!!! 

Adding these three equations together will give us another true equation.

Like this...

ax + a + ay + by + bx + b + c + cy + cx  =  x + 2x + 4x + 6y + y + 7     Then factor it like this...

a(x + 1 + y) + b(y + x + 1) + c(1 + y + x)  =  7x + 7y + 7      Divide through by  (x + y + 1) .

a + b + c  =  \(\frac{7x+7y+7}{x+y+1}\)

a + b + c  =  \(\frac{7(x+y+1)}{x+y+1}\)

a + b + c  =  7

*edit*

Also... to show that it produces a true equation when you add them together...let's say...

A = B  ,   C = D ,  and  E = F                          (And  A, B, C, D, E, and F  can be expressions.)

A + C + E  =  A + C + E      Now we can substitute  B  in for  A ,  D  in for  C , and  F  in for  E .

A + C + E  =  B + D + F

Here's one possibility

Here's a possibility  [ but maybe not the only one  !!! ]

We can solve this system

Asin (35pi/70) + B  = 120         note  .......Asin (35pi/70) =  Asin (pi/2)  = A

Asin (55pi/70) + B = 104

A  + B  = 120

A + .62349B= 104

Solving this system for A and B produces   A = 42.4956   B =  77.5044

The beginning temperature ≈ 77.504

Here's a graph : https://www.desmos.com/calculator/wzxfo0xmao

Note.....the points (35, 120)  and (55, 104) are on the graph

a)  FV=$546.54            Interest earned =$546.54 - $500 =$46.54

b) PV = $1,200             Interest earned =$1,616.83 - $1,200 =$416.83

c) Interest rate =7.5% compounded monthly       Interest earned =$1,381.02 - $1,250 =$131.02

d)  Timeframe =2,917.29 days, 0r =~97 1/4 months, or =~8 years. Int. earned =$2,025 - $875=$1,150

According to WolframAlpha.....as written, this isn't an identity........

I've been trying to find some other arrangement that might "work"  ....but I haven't hit on any, yet.....!!!!

1/cosx(tanx-1)=tanx + cotx/tanx-cotx

You have not used enough brackets, so your question is not clear.

This is what you have asked.  Is it what you really want?

\(\frac{1}{cosx(tanx-1)}=tanx + \frac{cotx}{tanx}-cotx\)

I always remember this formula as the square root of the sum of the difference of the x-coordinates squared and the difference of the y-coordinates squared.

For me, knowing how to say the formula in words is better than knowing that the distance formula is "the square root of x₁ minus x₂ all to 2nd power plus y₁ minus y₂ to the 2nd power. 

The same applies for the midpoint formula.

I remember it as the ordered pair of the average of the x-coordinates and y-coordinates.

We have that

a₁ + 4d  = 9     and

a₁ + 31d = -84       where a₁ is the first term  and d is the common difference between terms

Subtract both equations

- 27d  = 93      divide both sides by -27

d = - 93 / 27  = -31/ 9

And.....there are 18 terms between the 5th term and the  23rd  term......so we have....

9 + 18(-31 / 9)  =  9 - 62   =   -53   =   23rd term

Thank you soooooooo much, I will for sure watch them. <3

I never could remember the distance formula either.....until I finally understood where it came from!

It is -57

You don't want the "rate of return" on each investment since it is already stated in each case, i.e.,  6.5%, 7.8%, and 9.8%. What you want is the total worth of each investment at the end of its term:

a) $40,000 x  1.065^5=$54,803.47 Principal + interest compounded after 5 years.

b) $10,000 x 7.8% =$780 simple interest for 1 year

$780 x 5 =$3,900 simple interest for 5 years.

$10,000 + $3,900 =$13,900 principal + interest for 5 years.

c)  This is a bit more complicated. First, you have to convert the interest rate from compounded semi-annually to compounded monthly to match the monthly deposits of $2,500 each.

9.8 /200 =0.049 + 1 =1.049^1/6 =1.00800475 - 1 =0.00800475 x 12 =9.61% compounded monthly.

Then you have to use this formula to find the FV of the deposits:

FV=P{[1 + R]^N - 1/ R}

FV =2,500[1 + 0.00800475]^(5*12) -1 / (0.00800475)}

FV=$191,588.49 - This is the balance in the acct. after 5 years.

As you can see, there is no comparison to c) investment. You may have made a mistake in the monthly payment of $2,500. It may be $250 per month. If that is the case, then you just divide the last answer of $191,588.49 by 10 =$19,158.85.

If     w  =  \(c\cdot\frac{y}{c}-y\)     , then this is how to do it.

c  =  2 · 10⁶

y  =  2 · 10⁵

\(w^2\,=\,c\cdot\frac{y}{c}-y\)                                      Plug in the given values for  c  and  y .

\(w^2\,=\,2\cdot10^6\cdot\frac{2\cdot10^5}{2\cdot10^6}-2\cdot10^5 \\ w^2\,=\, 2\cdot10^6\cdot\frac{10^5}{10^6}-2\cdot10^5 \\ w^2  \,=\,   2\cdot10^6\cdot\frac{1}{10}-2\cdot10^5 \\ w^2  \,=\,   2\cdot10^5-2\cdot10^5 \\ w^2\,=\,0\\w\,\ \,=\,0\)              Reduce the fraction by  2 .

Or....if     \(w^2\,=\,\frac{c\cdot y}{c-y} \)     , then this is how to do it.

\(w^2\,=\,\frac{2\cdot10^6\cdot2\cdot10^5}{2\cdot10^6-2\cdot10^5} \,=\, \frac{4\cdot10^{11}}{20\cdot10^5-2\cdot10^5} \,=\,\frac{4\cdot10^{11}}{18\cdot10^5}\,=\,\frac{2\cdot10^6}{9}\\~\\ w\,=\,\pm\sqrt{\frac{2\cdot10^6}{9}}\,=\,\pm\frac{10^3\sqrt2}{3}\,=\,\pm\frac{1000\sqrt2}{3}\)

The second one is the same answer as Guest's.