All Answers 359404 Answers

Sqrt(169) = Thirteen

$$\small{\text{
$
\begin{array}{rcl}
f(x)=x+2 \\
f^{-1}(x) &=& ? \\
y & =& x+2 \quad | \quad x\leftrightarrow y \\
x & = & y+2 \\
y & = & x-2 \\
f^{-1}(x) & = & x-2 \\
\hline
\\
g(x)=\dfrac{x}{3} \\ \\
g^{-1}(x) &=& ? \\
y & =& \dfrac{x}{3} \quad | \quad x\leftrightarrow y \\ \\
x & = & \dfrac{y}{3} \\ \\
y & = & 3x \\
g^{-1}(x) & = & 3x \\
\hline
\end{array}
$
}}$$

$$\small{\text{
$
\begin{array}{rcccl}
f(19) & = & 19 + 2 & = & 21 \\
g(21) & = & \frac{21}{3} & = & 7 \\
f^{-1}(7) & = & 7-2 & = & 5 \\
f^{-1}(5) & = & 5-2 & = & 3 \\
g^{-1}(3) & = & 3*3 & = & 9 \\
f(9) & = & 9 + 2 & = & 11 \\
\end{array}
$
}}$$

$$f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))) = 11$$

1230 / 82  =  15

It can be shown that any non-prime integer can be formed by product of primes.......thus, in simplest terms, they help us "factor" numbers so that we can reduce fractions, etc.

So...putting everything in terms of % of water, we want the final mix to contain 70% water....note that pure water = 100%  = 1

So we have

10*.40 + 1x = (x + 10)*.70     simplify

4 + x = .7x + 7       subtract 4, .7x from both sides

3x = .3x                 divide both sides by .3

3/.3 = x = 10 litres

sec^2x (1 + cot^2x) =

1/ cos^2x (1  + cos^2x / sin^2x) =

1/ cos^2x +  1/sin^2x =

sec^2x + csc^2x

(Whether this is any more "simple" than the original expression is debatable...)

Note, oddly, that 1 + cot^x = csc^2x

So

sec^2x (csc^2x) =

sec^2x csc^2x

Thus the sum of the two functions, in this case = their products....!!!

Using.... ln a - ln b  = ln (a / b), we have

ln(x)^4-ln(x-9) =

ln[x^4 / (x - 9) ]

I get something a little different here...the angle between the ship's path and the line of sight is 28.5 degrees. CB is the adjacent side and we're looking for BA (the hypoteneuse, or line of sight distance).....so we have

cos 28.5 = 3.5 / BA

BA = 3.5 / cos 28.5  = about 3.98 km (or, 4 km, rounded)

Here's an illustration.......

Theta is a Greek letter. Here is all of the ones I know:

α β γ ε η θ ω π ξ

On the calculator it says 'error', but really there is no answer.

Yep. I mean that it does work for displaying the answer. Simply type a question on a scientific calculator, then see if this one shows the same thing.

Answer: 4 km 

12 k/m divided by 3  = 4 k/m. (60 minutes divided by 3 = 20 minutes)

It is one

It is one

Are you sure that is not meant to be a plus - it is from pythagoras theorem ?

To find out, replace x with -3:

Left side:     4x - 3 + 2x  =  4(-3) - 3 + 2(-3)  =  ?

Right side:   4 + x - 8  =  4 + (-3) - 8  =  ??

Does the answer for  ?  match the answer for  ?? 

If they match, -3 is an answer; if they don't match, then -3 isn't an answer.

I do it different from many people because i do not usually use algebra.

My method is very easy and logical.

Take a look at this post that I did recently.

When she enters the water, f(x) = 0.

0 = -4.9t² + 4t + 10

Solve for t by using the quadratic formula:  t =  [ -4 ± √(4² - 4 · -4.9 · 10) ] / [2 · -4.9 ]    --->  t = 1.89 sec.

Your question makes no sense to me.  Sorry. 

The area of the square is so large that it completely encloses rectangle R. So, let's just create a rectangle with one endpoint on the line  y = -3x and placed horizontally.

From a point (x, -3x), create a rectangle to the right and up. The distance from this point to the right end of rectangle R is: 8 - x; the distance from this point up to the top of rectangle R is:  12 - -3x).

The total area is:  (8 - x)(12 - -3x)  =  (8 - x)(12 + 3x)  =  96 + 12x - 3x².

The first derivative is:  12 - 6x

Setting this equal to zero:  12 - 6x  = 0  --->   12  =  6x   --->   2 = x   

If  x = 2,  y = -6

The horizontal length is 8 - 2  =  6, the vertical length is 12 - -6  =  18     --->   Area  =  108

ThANKS  ROSALA,

I am glad that YOU are back AND that you have brought you FUNNIES with you     

$$1400 \div 35000\\\\
=\frac{1400}{35000}\\\\
=\frac{14}{350} \qquad \mbox{I divided both by 100}\\\\
=\frac{2}{50} \qquad \mbox{I divided both by 7}\\\\\
=\frac{1}{25} \qquad \mbox{I divided both by 2}\\\\\
$so if there are common factors you simplify them as fractions$$$

Do you mean

$$6.\bar 9$$

That dot can be in the middle, it makes no difference.  I would like it better if it were in the middle. :)

The notation for this is slightly different from country to country but it means

6.999999999999999999999999999999999999999999999999      and the nines keep going for ever.   

$$\\(K^{0.75}/L^{0.75})/2 = (3L^{0.25}/K^{0.25})/10\\\\
5(K^{0.75}/L^{0.75}) = 3L^{0.25}/K^{0.25}\\\\
5(K/L)^{0.75} = 3(L/K)^{0.25}\\\\
5(K/L)^{0.75}= 3(K/L)^{-0.25}\\\\
$Let $ R=K/L\\\\
5(R^{0.75}) = 3(R)^{-0.25}\\\\
(R^{0.75})/(R)^{-0.25} = 3/5\\\\
R^1=3/5\\
r=3/5$$

I don't know about the minimize bit.

I just get the ratio of K to L  is  3:5

 square root of 81 is sqr$${\sqrt{{\mathtt{81}}}}$$ ,9*9=81 , therefore the answer is 9

Yes that proves it - I'm a great teacher!!     LOL