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Mmm good question...

Squared numbers must end in  1,4,9,6,5 or 0

So if your whole number ends in 2,3,7 or 8 then the square root will have to be irrational.

I have not answered your question. I have only looked at the most obvious exclusions.  

Perhaps if you give some specific examples I could tell you what other logic I would use. ://

Maybe someone else would like weigh in?

Oh OK. It's simple.

Plug this decimal into the fraction

0.6744302788099833

x + 9(x+4)=

x+9x+36=

10x+36

.1=1/10

.01=1/100

it's pretty simple, just plug your decimal into the fraction

x^2+4x+9x+36 

x^2+13x+36 

x + 9 (x+4)

x  +  9x  +  36

10x  + 36

2(5x  +  18)

Solve for a:
a^3 + 1/a^3 = 6

Bring a^3 + 1/a^3 together using the common denominator a^3:
(a^6 + 1)/a^3 = 6

Multiply both sides by a^3:
a^6 + 1 = 6 a^3

Subtract 6 a^3 from both sides:
a^6 - 6 a^3 + 1 = 0

Substitute x = a^3:
x^2 - 6 x + 1 = 0

Subtract 1 from both sides:
x^2 - 6 x = -1

Add 9 to both sides:
x^2 - 6 x + 9 = 8

Write the left hand side as a square:
(x - 3)^2 = 8

Take the square root of both sides:
x - 3 = 2 sqrt(2) or x - 3 = -2 sqrt(2)

Add 3 to both sides:
x = 3 + 2 sqrt(2) or x - 3 = -2 sqrt(2)

Substitute back for x = a^3:
a^3 = 3 + 2 sqrt(2) or x - 3 = -2 sqrt(2)

Taking cube roots gives (3 + 2 sqrt(2))^(1/3) times the third roots of unity:
a = -(-3 - 2 sqrt(2))^(1/3) or a = (3 + 2 sqrt(2))^(1/3) or a = (-1)^(2/3) (3 + 2 sqrt(2))^(1/3) or x - 3 = -2 sqrt(2)

Add 3 to both sides:
a = -(-3 - 2 sqrt(2))^(1/3) or a = (3 + 2 sqrt(2))^(1/3) or a = (-1)^(2/3) (3 + 2 sqrt(2))^(1/3) or x = 3 - 2 sqrt(2)

Substitute back for x = a^3:
a = -(-3 - 2 sqrt(2))^(1/3) or a = (3 + 2 sqrt(2))^(1/3) or a = (-1)^(2/3) (3 + 2 sqrt(2))^(1/3) or a^3 = 3 - 2 sqrt(2)

Taking cube roots gives (3 - 2 sqrt(2))^(1/3) times the third roots of unity:
a = -(-3 - 2 sqrt(2))^(1/3)   or   a = (3 + 2 sqrt(2))^(1/3)   or   a = (-1)^(2/3) (3 + 2 sqrt(2))^(1/3)   or a = (3 - 2 sqrt(2))^(1/3)   or   a = (-1)^(2/3) (3 - 2 sqrt(2))^(1/3)   or   a = -(2 sqrt(2) - 3)^(1/3)

A = gr      divide both sides by r

g =A / r

3 / [3+13]  x   6,816 =1,278 are pixies.

13 / [3+13]  x  6,816 =5,538 are fairies.

17)  Bolean contour mapping to solve probablility questions.     (29/9/17)

1)   We want the ratio of width to height to be the same. If we multiply the height by 2, we must

      multiply the width by 2. If we multiply the height by  x , we must multiply the width by  x .

      The number that we multiply the height and width by is the scale factor.

\(\frac{\text{old width}}{\text{old height}}\,=\,\frac{\text{new width}}{\text{new height}} \\~\\ \frac{10}{8}\,=\,\frac{\text{new width}}{3.5} \\~\\ 3.5\,*\,\frac{10}{8}\,=\,\text{new width} \\~\\ 4.375\,=\,\text{new width, in inches}\)          Plug in the information from the problem.

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2)   This is done the same as the last question.

\(\frac{\text{old width}}{\text{old height}}=\frac{\text{new width}}{\text{new height}} \)          Plug in the information from the problem.

\(\frac{14}{10}=\frac{\text{new width}}{2}\)          Can you finish it from here?  

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3)   First let's convert 3 feet into inches.   1 ft  =  12 in   →   3 ft  =  36 in

Now let's call the unknown scale factor  " s " .

4.5s  =  36      Divide both sides by  4.5 .

     s  =  8

So each side of the triangle is multiplied by  8  .

The base of the larger triangle will then   =   (8 * 6) inches   =   48 inches   =   4 feet

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4)   If it takes 0.03 inches to make 1 mile on the map, then it will take 0.03 * 120 inches to make 120

      miles on the map...which is 3.6 inches.

I just noticed something that is kind of interesting!!

You could also have factored it as    (7m)(1 + 2m)(1 – 2m)

So...        ( – 7m)(2m + 1)(2m – 1)  =  (7m)(1 + 2m)(1 - 2m)

which means....                 – (a + b)(a – b)  =  (b + a)(b – a)      

Yeaa!! I got it right!!!!!!!!!!!

Very nice, hectictar.....!!!!!

Let  a  be the number of species A ants he had to begin with.

Let  b  be the number of species B ants he had to begin with.

On day 0 there are  a + b ants.

On day 1 there are  2a + 3b  ants.

On day 2 there are  2 * 2a + 3 * 3b  ants.

On day 3 there are  2 * 2 * 2 * a + 3 * 3 * 3 * b  ants.

On day 4 there are  2⁴ * a + 3⁴ * b  ants.

On day n there are  2ⁿ * a + 3ⁿ * b

So...

On day 5 there are 2⁵ * a + 3⁵ * b

a + b  =  30     →     b = 30 - a

2⁵ * a  + 3⁵ * b  =  3281

2⁵ * a + 3⁵ * (30 - a)  =  3281

2⁵a + 3⁵ * 30 - 3⁵a  =  3281

2⁵a - 3⁵a  =  3281 - 3⁵ * 30

a(2⁵ - 3⁵)  =  3281 - 3⁵ * 30

a  =  (3281 - 3⁵ * 30) / (2⁵ - 3⁵)

a  =  19

So...he started with  19  ants of species A.

After  5 days, he will have  2⁵ * 19  =  608 ants of species A.

Call the amount invested at 4%  = .04 =   = x

So   the amount invested at   3.5%  =  .035  =  5000 - x

And     Amount * Interest Rate  = Total Interest   .....so...

Amount at 4%    +   Amount at 3.5%  =  190          and we have that

(x) (.04)   +  ( 5000 - x) (.035)  = 190      simplify

.04x  +  175  - .035x  =   190          combine like terms     

.005x +  175   = 190                   subtract 175 from both sides

.005x  =   15                divide both sides by  .005

x  = $3000   = amount   invested at  4%

5000 - x  =    5000  - 3000  = $2000  = amount invested at 3.5% 

A ratio is the relationship (sometimes denoted as the quotient) of 2 separate quantities. 

Therefore, since the relationship is the number of seventh graders to the number of eighth graders.

\(=\frac{24\hspace{1mm}\text{seventh graders}}{30\hspace{1mm}\text{eighth graders}}\)

Just like normal fractions, one should simplify such that the greatest common factor of both the numerator and denominator is 1.

\(\frac{24\hspace{1mm}\text{seventh graders}}{30\hspace{1mm}\text{eighth graders}}\div\frac{6}{6}=\frac{4\hspace{1mm}\text{seventh graders}}{5\hspace{1mm}\text{eighth graders}}\)

We want to know... how many cells are there in a cubic​ centimeter?

That is.....how many  200 cubic micrometers  are there in a cubic centimeter?

We want to know...what is   \(\frac{1\,\text{ cubic centimeter}}{200\,\text{ cubic micrometers}}\)    ?

First let's convert 200 cubic micrometers into cubic centimeters.

1 meter  =  1 · 10² centimeters  =  1 · 10⁶ micrometers

So.....

\(\frac{1\,\text{ cubic centimeter}}{200\,\text{ cubic micrometers}}\,=\,\frac{1\,\text{ cubic centimeter}}{2 \cdot 10^{-10}\,\text{ cubic centimeters}}\,=\,\frac{1}{2\cdot10^{-10}}\,=\,5\cdot10^9\)

Note that the sum of the first 21 integers  is   21 * 22 /2  =  231...this isn't large enough

And the sum of the first  22 integers  =   22 * 23 / 2  =   253

So    253 - 241  =  12 = omitted sum.....  but  the sum of two consecutive integers must be odd

And......the sum of the first 23 integers is  23 * 24 / 2  = 276

So.......276 - 241   =  35    = omitted sum

So....the   consecutive integers omitted must be  17  and 18

So....... the smallest value of n  is   23

Thanks, X^2....I neglected to spot that 1 - 4m^2  could be factored further as  (1 -2m) (1 + 2m)  !!!!!

\(7m-28m^3\)

The first step is to figure out the GCF. Of course, Cphill has already figured it out and has provided an explanation that it is 7. However, you can also divide by -7, which is what I will do here.

\(-7m(-1+4m^2)=-7m(4m^2-1)\)

4m^2-1 happens to be a difference of 2 squares, though! We must factorize that! 

No more progress can be made.

Therefore, the final factorization is \(-7m(2m+1)(2m-1)\)

7m - 28m^3           

The greatest common factor   between   7 and  28   = 7

The greatest common factor between m and m^3 =   m

So.... the greatest common factor  ( GCF )  = 7m

So we will have

7m ( a  -  b)

To  find out  what  " a '  will be....divide the first term of the given expression , 7m,  by the GCF

So      7m  /   7m    = 1

To find out what  "b" will be......divide the second term of the given expression, 28m^3,  by the GCF

So       28m^3  / 7m  =      4m^2

So......the complete factorization is

7m  ( 1  - 4m^2)