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Open Windows charcter map     ....   copy from there .....

I think you can use text symbols that you can copy and paste to use in word file and even anywhere you want.

No problem. The technique with choose binomials is very helpful in geometric counting/probability.

The theorem used is called the power of a point theorem. More about that here: https://en.wikipedia.org/wiki/Power_of_a_point

More elaborate solution:

Let the two numbers be x and y, where x is smaller.

By the first sentence, 7x=3y --> x=3/7y.

We also know that xy=189 --> (3/7y)(y)=189 --> 3/7y^2=189 --> y^2=441 --> y=21,-21.

When y=21, we have 7x=3*21 --> x=9

When y=-21, we have 7x=3*-21 --> x=-9

So our solutions are (x,y)=(21,9) and (-21,-9)

The smallest possible number is then -9

\(10^2\pi-6^2\pi=100\pi-36\pi=\boxed{64\pi}\)

:)

Thank you

See this for squares: http://puzzles.nigelcoldwell.co.uk/twentyseven.htm

For rectangles: 

We can choose the two verticals in \(\binom{8+1}{2}=36\) ways, and by symmetry, it is the same for horizontals. So 36^2=1296.

Start by drawing a diagram. What do you observe?

Some properties you might want to know: Vertical angles are equal, Angles on a line add up to 180 degrees, Angles in a triangle add up to 180 degrees.

By the binomial theorem, this term is \(\binom{11}{2}x^93^2=55*9x^9=495x^9\rightarrow\boxed{495}\)

\(\left(\frac{11}{2}\right)^2=\frac{121}{4}\rightarrow (t+\frac{11}{2})^2=\boxed{t^2+11t+\frac{121}{4}}\)

\(13^2\pi-12^2\pi=169\pi-144\pi=25\pi\rightarrow |\sqrt{\frac{25\pi}{\pi}}|=\boxed{5}\)

The distributive property.

i'm sorry about that, that was a typo I had

LCM of (180, 675)  = 2700
GCD of (180, 675)  = 45

that didn't work

tan = sin/cos       so wherever sin = cos    tan = 1

  pi/4     and    5pi/4         (45^o   and  225^o )

Who gave the thumbs down for this?  The solution is accurate and, if I do say so myself, clear and easy for young students to understand how to achieve it.  So why the thumbs down?  It isn't my fault that the problem describes a triangle that can't exist. 

I'm going back to answer questions in the Knowledge Base on eBay.  No hateful trolls giving me unhelpfuls.  They appreciate help over there, where, incidentally, I was accepted into the mentor group and have 18,378 Helpfuls as of this writing  It might be immodest for me to toot my own horn, but if I don't, who else will?  And, as the great Muhammad Ali said, It ain't bragging if you can do it. 

_.

Amounts of detergent:

s = small

l = large =  2s

m = medium  =  .8 l  = 1.6s

Assign the unit cost of

small                         = 1  per unit

   then

m = 1.5s / 1.6s         = .9375 per unit

l  =   (1.3)(1.5s) / 2s  = .975  per unit

so the cost per unit (best to worst)        medium     large    small   

Correct! Thanks!

Identify the value of x and the length of each chord. 

The "Intersecting Chords Theorem" states 

      The intersecting chords theorem or just the chord theorem is a statement in elementary geometry   

      that describes a relation of the four line segments created by two intersecting chords within a circle.   

      It states that the products of the lengths of the line segments on each chord are equal.   

I hadn't known this before.  A tip of the hat to jugoslav. 

_.

1 /  (4/3)   =    3/4

3/ 4  *   4/3    = 1 

What is the reciprocal of 4/3   

I found the following definition of reciprocal: 

     a mathematical expression or function so related to another that their product

     is one; the quantity obtained by dividing the number one by a given quantity.  

I went huh and blinked a couple of times.  I prefer the earlier guest's answer, i.e., Just flip it over. 

_.

**** edit ****

2\9-(-4\3+7\8-2) =

2/9 + 4/3 -7/8 + 2 =

17/72   +   96/72    -   63/72 + 144/72 =

(17  +  96  - 63  + 144) / 72