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.5  1 is a whole  1.5 is 1 whole and a half 

$$\small{\text{Compute the value of $22^2 + 6 \cdot 22 + 9$.}}$$

$$\small{\text{$
\begin{array}{rcl}
22^2 +6\cdot 22 + 9 &=&(22+3)^2 = 25^2 = 625
\end{array}
$}}$$

I have only just now had a chance to look at your answer Heureka.

I really like it.      

I have added this thread address to our "Great Answers to Learn From" sticky thread.  :)

A half of a whole is a half   

$$\\2-\frac{2}{x}\div 3-\frac{1}{x}\\\\
=2-\frac{2}{x}\div \frac{3}{1}-\frac{1}{x}\\\\
=2-\frac{2}{x}\times \frac{1}{3}-\frac{1}{x}\\\\
=2-\frac{2}{3x}-\frac{1}{x}\\\\
=2-(\frac{2}{3x}+\frac{3}{3x})\\\\
=2-\frac{2+3}{3x}\\\\
=2-\frac{5}{3x}\\\\
or
=\frac{6x-5}{3x}$$

6-5/x$${\mathtt{6}}{\mathtt{\,-\,}}{\frac{{\mathtt{5}}}{{\mathtt{x}}}} = {\mathtt{6}}{\mathtt{\,-\,}}{\frac{{\mathtt{5}}}{ABC}}$$

aaso an $$\pm 1\quad and \quad \pm121$$

tan x + cot x  =

sinx/cosx + cosx/sinx  =   [get a common denominator]

[sin^2x + cos^2x] / [sinx*cosx] =

1/ [sin x * cosx]       and this equals the RHS

3x/[x^2 - 4] + 6/[x + 2]   =

3x / [(x + 2)(x -2)] +  6/[x + 2]        get a common denomonator......

3x/[(x + 2)(x -2)] + 6[x -2] / [(x + 2)(x - 2)]   =

[3x + 6x - 12] /  [(x + 2) (x -2)]  =

[9x - 12] / [ (x + 2) ( x- 2)]      and x cannot equal 2 or -2  [those make the denominator = 0 ]

Let R be her "normal" rate.....and R - 10 be her "reduced" rate......

And R*Time   = Distance......so......since the distances are equal.....we have.....

R(1/2)hr      =   (R - 10)(1/2  + 1/6) hr  .....  [10 min = 1/6 hr]

(1/2)R  =  (R -10)(2/3)

(1/2)R  =  (2/3)R - 20/3    rearrange

20/3  = (2/3 - 1/2)R

20/3  = (1/6)R     multiply both sides by 6

120/3   = R  = 40   and this is her normal rate in mph

And, her "reduced" rate is [40 - 10]   = 30 mph

And the distance to work is (1/2)h(40)m/h  = 20 miles

  x + y  =  10     --->    -x - y  =  -10

2x + y  =  13     --->   2x + y  =  13

Add down the columns:  x  =  3

Substituting into the first equation:  y = 7

x² - y²  =  (3)² - (7)² =  9 - 49  =  -40

"pi" or "pie"

pi is an infinitely long decimal; it never stops.

Let p = amount of money that Pat has.

Let b = amount of money that Bob has.

If Pat gives Bob a dollar, then Pat will have p - 1 dollars and Bob will have b + 1 dollars.

Since Bob will have twice as much money as Pat has:  b + 1  =  2(p - 1).

If Bob gives Pat a dollar, then Pat will have p + 1 dollars and Bob will have b - 1 dollars.

Since they will then have the same amount of money:  p + 1  =  b - 1.

Putting these two equations together:

b + 1  =  2(p - 1)   --->   b + 1  =  2p - 2   --->   3  =  2p - b

b - 1  -  p + 1         --->                                       -2  =    p - b

2p - b  =  3    --->                     2p  - b  =  3

  p - b  =  -2   --->   x -1   --->   -p + b  =  2

Adding down the columns:                  p  = 5

Substituting back into an equation:     b  =  7

If you take the original 48 eggs and divide by the number of chickens,

you get 48 ÷ 6  = 8 eggs per chicken.

Dividing this by the number of days, you get 8 ÷ 4  = 2 eggs per chicken per day.

If each chicken can lay 2 eggs each day, 10 chickens can lay 20 eggs per day.

To get a total of 100 eggs, you need 100 ÷ 20  = 5 days. 

-10xyz · -y²z

Negative times negative = positive

10xyz · y²z

=  10xy³z²

(when multiplying expressions with the same base, add the exponents)

If 12 bears eat 5 pounds of honey in one day, the each average 5/12 of a pound per day.

Therefore for 20 bears in three days:  20 x (5/12) x 3  =  25 pounds

18/-12  =  -3/2

(a - b)³/(a - b)²  =  (a - b)

Putting these together:  -3(a - b)/2

y⁰ = 1

--->   x³ · 1 · x^-7  =  x³ ·  x^-7  =  x^-4

(When you multiply expressions with the same base, add the exponents.)

I would do it in two steps:

First:  multiply the numerator and denominator by the common denominator:  

             (√(x + h))·(√x)

Giving:

[ 3√(x + h) + 3√x ] / [ h·√(x + h)·√x ]

and factor out the 3:  3[√(x + h) + √x ] / [ h·√(x + h)·√x ]

Now, multiply the numerator and denominator of this fraction by the conjugate of the numerator:

              √(x + h) - √x 

Numerator:  the product of 3[√(x + h) + √x ] times [ √(x + h) - √x  ]  

                    =  3[ (x + h) - x ]  =  3h

Denominator:  the product of [ h·√(x + h)·√x ] times [ √(x + h) - √x  ]  

                    =  h(x+h)√x  - h√(x + h)·x    =  h[ (x+h)√x  - √(x + h)·x ]

                    =  h[ √x(x + h) - x√(x + h) ]

Cancelling the h from both the numerator and the denominator gives you their answer.

Use FOIL:  First, Outside, Inside, Last:

(x + 5)²  =  (x)(x) + (5)(x) + (5)(x) + (5)(5)

               =  x² + 5x + 5x + 25

               =  x² + 10x + 25

There is a calc on the home page for that. :)

If you know combinations:

= C(5,0)(3b)⁵(0.5)⁰ + C(5,1)(3b)⁴(0.5)¹ + C(5,2)(3b)³(0.5)² + C(5,3)(3b)²(0.5)³ 

                                                                  + C(5,4)(3b)¹(0.5)⁴ + C(5,5)(3b)⁰(0.5)⁵ 

=  1(3b)⁵(0.5)⁰ + 5(3b)⁴(0.5)¹ + 10(3b)³(0.5)² + 10(3b)²(0.5)³ 

                                                                  + 5(3b)¹(0.5)⁴ + 1(3b)⁰(0.5)⁵ 

=  1(243b⁵)(1) + 5(81b⁴)(0.5) + 10(27b³)(0.25) + 10(9b²)(0.125)

                                                                  + 5(3b)(0.0625) + 1(1)(0.03125)

=  243b³ + 202.5b⁴ + 67.5b³ + 11.25b² + 0.9375b + 0.03125

If you don't know the rule of combinations, use the 5th row of Pascal's triangle.

Sorry anon - That is not correct.

you need to use the binomial expansion

I am on my phone at the moment so it is all to difficult for me right now.

if no one gives a timely answer why not Google  binomial expansion and watch a you tube presentation.

it would most likely be easier to understand anyway. :)

Ln is log base e

It turns through 2pi x 1.5   =  3pi revolutions in one minute......so,  in 30 seconds, it turns through 1/2 of this  =  (3pi/2)

And the distance - (s) - moved by the point is  

S = r(Θ)   =  (30cm)(3pi/2)   = [90pi/2] cm = [45pi ] cm =  about  141.37 cm