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Thanks so much!!!! You're the best

Let's first write this as

y = 16 / [ 5 + x ]

We need to find the inverse ....the idea is to isolate x and then "exchange" x and y

Multiply both sides by [ 5 + x ]

[ 5 + x ] y =  16     now....divide both sides by y

5 + x  = 16/y        subtract  5 from both sides

x =  16/y  - 5    "  exchange " x and y

y = 16/x - 5       get a common denominator on the right

y =  16/x  -  5x / x

y =  [ 16 - 5x ] / x      and for y, write f^-1(x)

f^-1(x) =  [ 16 - 5x ] / x

Now....we want to first find  [ f^-1(2 ) ]

This means that we want to put 2 into the inverse and evaluate it.....so we have

[ 16 -  5(2) ] / 2  =   6/2  = 3

So..... [ f^-1(2 ) ] =  3

Now we want to find   [ f^-1(2 ) ] ^-2    which is just [ 3 ]^-2

Remember that a^-m  =  1 / a^m

Therefore..... 3^-2  =  1 / 3²   =   1 / 9

So......[ f^-1(2 ) ] ^-2 =   1 / 9

And that's it..!!!!

Let y be age of youngest.

y + (y + 1) + 3y = 21

5y + 1 = 21

5 y = 20

y = 4. Age of youngest

y + 1 = 5. Age of middle child

3y = 12. Age of oldest child.

Thank you!!

Use v^2 = u^2 + 2as

v = 0 m/s. Final velocity

u = 80000/3600 m/s. Initial velocity

a = -7.5 m/s^2.  Acceleration (-ve sign indicating deceleration)

s is distance travelled in metres.

s = (80000/3600)^2/(2*7.5) m → 32.9 m

.

We have:

 2(3x+c) + 7 = 3(2x+7) + c

6x + 2c + 7 = 6 x + 21 + c

2c + 7 = 21 + c

c = 14

.

3n + 4001 + 3n + 4001 + 3n + 4001 =

 (3n + 4001) + (3n + 4001) + ( 3n + 4001) =

3 ( 3n + 4001)  =

9n + 12003

a^2 - 6ab - 7b^2

(a  - 7b) (a + b)

14e - 7f  =  28

 7e +   f =   50 

Note that if we divide the top equation through by 7, we have

2e - f = 4      add this to the second equation  and we get

9e = 54        divide both sides by 9

e = 54/9  =  6

And using the second equation to find f, we have

7(6) + f  = 50

42 + f  =  50       subtract 42 from both sides

f  =  8

y^2 +  2y  + 12x + 25 = 0     

Subtract y^2 + 2y + 25 from both sides

12x = -y^2 - 2y -25        divide both sides by 12

x = (-1/12)y^2 - (1/6)y -  25/12      this is the general form

y^2 +  2y  + 12x + 25 = 0

Now....let's find the vertex form

Subtract   12x, 5 from both sides

y^2 + 2y   =    -12x  -  25

Complete the square on y

Take (1/2) of 2  = 1....square this  = 1^2  = 1 and add to both sides

y^2 + 2y + 1  =  -12x  - 25 + 1

y^2 + 2y + 1  =  -12x  - 24     factor both sides

(y + 1)^2  =  -12(x + 2)          

-12(x + 2) = (y + 1)^2

And this is in the vertex form      4p(x - h) = (y - k)^2

The real reason is that "thermal radiation", in other words, HEAT, can travel through the vacuum of space just like light and ALL electromagnetic radiation, from gamma rays, x-rays, radio waves and ALL visible and invisible electromagnetic radiation.

2150% (2150 percents)

Considering that 1 is equal to 100 percent and 0.5 is 50 percent.

----*edit*-----

After thinking about this some more...I think this is a much better and simpler way to put it...

Jenny's number  =  2 * n²             , where  n  is an integer.

...I guess I'll leave the old answer there... even though I REALLY don't like it anymore    haha

---------------

The prime factorization of Jenny's favorite positive integer must contain an odd number of 2's and an even number of anything other than 2.

For example, it could be..

2 * 3 * 3  =  18   ,

2 * 2 * 2 * 5 * 5  =  200   ,

2 * 7 * 7 * 3 * 3  =  882   , or just

2  =  2  .

If it satisfies this condition, then the square root of that number can always be written like this..

\(\sqrt2*\sqrt{\text{any positive integer}^2} \\~\\ \sqrt2*\text{any positive integer}\)

And multiplying, or dividing, this by \(\sqrt2\)  will result in an integer.

True.

Not in general....

For instance, let x = 3 and y = 4

Then

x^y  =  3^4  =  81

But

y^x  =  4^3  =  64 

1657k+1162 =1162m + 453. By simple iteration:

k = 15 and  m =22. Therefore, the smallest n =1657*15+1162 =26,017

Since the LCM of 1657 and 1162 =1,925,434, therefore:

n = 1,925,434D + 26,017, where D =0, 1, 2, 3......etc.

*editing

We can rewrite the modular equations as:

\(n-1162=1657k,\\ n-453=1162m.\)

Now we are able to remove n from the equation:

\(1657k+1162=1162m+453,\)

we can split the first term to tidy things up:

\(1162k+495k+1162-1162m=453\Rightarrow\\ 1162(k+1-m)+495k=453\Rightarrow\\ 1162(k+1-m)+495k-495=453-495\Rightarrow\\ 1162(k+1-m)+495(k-1)=-42.\)

The first term is even and the outcome is even, hence the second term must be even as well. We conclude that 495(k-1) is a multiple of 10. In order to get a negative number (k+1-m) must be negative as well. Since the second term doesn't contribute to the last digit of the sum we know the last digit of the first term must also be 2. As a result (k+1-m) must have a last digit equal to 1 or 6.

Energy :)

As long as this is in a right triangle, and the opposite and adjacent sides are not the hypotenuse...

tan( angle )  =  opposite / adjacent

                                                               Plug in the values we know for opposite and adjacent.

tan( angle )  =  1290 / 48

                                                               Take the inverse tangent of both sides.

angle  =  arctan( 1290 / 48 )

                                                               Put this into a calculator to get...

angle  ≈  87.869º

I can simplify the elements but the question does not make any sense.

Α= √(√81) +3√8:√2+8√3:(1+√3)/(3+√3) 

\(Α=\sqrt9 +3*2\sqrt2:√2+8√3:\frac{(1+√3)}{(3+√3) }\\ Α=3 +6\sqrt2:\sqrt2+8\sqrt3:\frac{(1+\sqrt3)}{(3+\sqrt3)} \times \frac{(3-\sqrt3) }{(3-\sqrt3) }\\ Α=3 +6\sqrt2:\sqrt2+8\sqrt3:\frac{3-\sqrt3+3\sqrt3-3}{9-3}\\ Α=3 +6\sqrt2:\sqrt2+8\sqrt3:\frac{\sqrt3}{3}\\\)

Hi Docphys   

Welcome to web2.0calc forum  :))      

Sorry your post got temporarily blanked.    That happens sometimes :/     

This was answered here: https://web2.0calc.com/questions/factor-this-expression 

Here's another way to do that one....

2x² + 7x + 5

Multiply the  2  and the  5  together to get  10  .

What two numbers multiply to  10  and add to  7  ? →   +2   and   +5

So, we can split the middle term like this:

2x² + 2x + 5x + 5              Notice here how if we combined like terms, we'd get  7x  again.

                                          Factor  2x  out of the first two terms.

2x(x + 1) + 5x + 5

                                          Factor  5  out of the last two terms.

2x(x + 1) + 5(x + 1)

                                          Factor  (x + 1)  out of both remaining terms.

(x + 1)(2x + 5)

A rectangular sheet of paper measures 12 inches by 9 inches.

One corner is folded onto the diagonally opposite corner and the paper is creased.

What is the length in inches of the crease? Express your answer as a mixed number.

Let a = 12 inches
Let b = 9 inches
Let x = crease = ?

\(\begin{array}{|rcll|} \hline x &=& \frac{b}{a}\cdot \sqrt{a^2+b^2} \\ &=& \frac{9}{12}\cdot \sqrt{12^2+9^2} \\ &=& \frac{3}{4}\cdot \sqrt{225} \\ &=& \frac{3}{4}\cdot 15 \\ &=& \frac{45}{4} \\ &=& 11 \frac14\ \text{ inches}\\ \hline \end{array}\)

\(\begin{array}{|lrcll|} \hline (1) & b^2 &=& p\cdot a \\ & p &=& \frac{b^2}{a} \\\\ (2) & x^2 &=& p\cdot(p+a) \\ & x^2 &=& \frac{b^2}{a} \cdot(\frac{b^2}{a}+a) \\ & x^2 &=& \frac{b^2}{a} \cdot \frac{a}{a} \cdot(\frac{b^2}{a}+a) \\ & x^2 &=& \frac{b^2}{a^2} \cdot a \cdot(\frac{b^2}{a}+a) \\ & x^2 &=& \frac{b^2}{a^2} \cdot(b^2+a^2) \\ & x^2 &=& \frac{b^2}{a^2} \cdot(a^2+b^2) \\ & \mathbf{x} & \mathbf{=} & \mathbf{ \frac{b}{a} \cdot \sqrt{a^2+b^2} } \\ \hline \end{array}\)