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When the fraction 1/288 is expressed in base 12,

is it terminating or repeating? Explain.

\(\begin{array}{|rcll|} \hline \frac{1}{288} &=& a\cdot 12^{-1} + b\cdot 12^{-2}+ c\cdot 12^{-3}+ d\cdot 12^{-4}+ e\cdot 12^{-5} \dots \quad & | \quad \cdot 12 \\ \frac{12}{288}=\frac{1}{24} &=& a + \underbrace{ b\cdot 12^{-1}+ c\cdot 12^{-2}+ d\cdot 12^{-3}+ e\cdot 12^{-4} \dots }_{\text{decimal places}=\frac{1}{24}}\\ integer(\frac{1}{24}) &=& a \quad \Rightarrow \quad a = 0\\\\ \frac{1}{24} -a &=& b\cdot 12^{-1}+ c\cdot 12^{-2}+ d\cdot 12^{-3}+ e\cdot 12^{-4} \dots \\ \frac{1}{24} -0 &=& b\cdot 12^{-1}+ c\cdot 12^{-2}+ d\cdot 12^{-3}+ e\cdot 12^{-4} \dots \\ \frac{1}{24} &=& b\cdot 12^{-1}+ c\cdot 12^{-2}+ d\cdot 12^{-3}+ e\cdot 12^{-4} \dots \quad & | \quad \cdot 12 \\ \frac{12}{24}=\frac{1}{2} &=& b + \underbrace{ c\cdot 12^{-1}+ d\cdot 12^{-2}+ e\cdot 12^{-3} \dots }_{\text{decimal places}=\frac{1}{2}}\\ integer(\frac{1}{2}) &=& b \quad \Rightarrow \quad b = 0\\\\ \frac{1}{2}-b &=& c\cdot 12^{-1}+ d\cdot 12^{-2}+ e\cdot 12^{-3} \dots \\ \frac{1}{2}-0 &=& c\cdot 12^{-1}+ d\cdot 12^{-2}+ e\cdot 12^{-3} \dots \\ \frac{1}{2} &=& c\cdot 12^{-1}+ d\cdot 12^{-2}+ e\cdot 12^{-3} \dots \quad & | \quad \cdot 12 \\ \frac{12}{2}= 6.0000000\dots &=& c + \underbrace{ d\cdot 12^{-1}+ e\cdot 12^{-2} \dots }_{\text{decimal places}=0}\\ integer(6.0000000\dots) &=& c \quad \Rightarrow \quad c = 6 \\\\ \frac{1}{288} = 0.006_{12}\\ \hline \end{array} \)

That is really cluey guest.  I have not done a problem like this in yonks.

Here is another way to look at it.

\(\frac{1}{288}\;base \;10 = \frac{1}{2*10^2+8*10+8}\;\;base 10\)

But we want to try and convert this to base 12

288 is much bigger than 12 but how does it compare to 12^2

288=2*144=2*12^2

so

Since 288 is a multiple of a power of 12 then 1/288    MUST   be a terminating decimal in base 12

END OF QUESTION    

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So what it it???

\(\frac{1}{288_{10}}=\frac{1}{2*12^2}\;(base10)\;\;=\frac{1}{2}of 0.01 \;\;(base\;12) = 0.006_{12} \quad\text{(because half of 12 is 6)}\\ or\\ 12^3=1728 \;\;and\;\; 1728\div 288=6\;\;\\so\\ \frac{1}{288_{10}}=\frac{1}{\frac{12^3}{6}}\;\;base10=\frac{6}{12^3}\;\;base10\;\;=0.006_{12} \)

The product of two integers is -144.
The sum of the integers is -7.
What are the two integers?

\(\begin{array}{|rcll|} \hline n_1 + n_2 &=& -7 \\ n_2 &=& -7 - n_1 \\\\ n_1\cdot n_2 &=& -144 \\ n_1\cdot (-7 - n_1) &=& -144 \\ -7\cdot n_1 - n_1^2 &=& -144 \quad & | \quad \cdot(-1)\\ 7\cdot n_1 + n_1^2 &=& 144 \quad & | \quad -144\\ 7\cdot n_1 + n_1^2 -144 &=& 0\\ n_1^2 + 7 n_1 -144 &=& 0\\ n_{1,2} &=& \frac{-7\pm\sqrt{(-7)^2-4\cdot(-144)} }{2} \\ n_{1,2} &=& \frac{-7\pm\sqrt{49+4\cdot 144} }{2} \\ n_{1,2} &=& \frac{-7\pm\sqrt{625} }{2} \quad & | \quad \sqrt{625} = 25\\ n_{1,2} &=& \frac{-7\pm 25 }{2}\\ n_1 = \frac{-7+25}{2} &\text{or}& n_1 = \frac{-7-25}{2} \\ n_1 = 9 &\text{or}& n_1 = -16 \\\\ n_2 = -7 - n_1 &\text{or}& n_2 = -7 - n_1 \\ n_2 = -7 - 9 &\text{or}& n_2 = -7 - (-16) \\ n_2 = -16 &\text{or}& n_2 = -7 +16 \\ & & n_2 =9 \\ \hline \end{array}\)

The two integers are 9 and -16

The question is incomplete here. The statement provided by Ashley and Pam don't have a connection factor. 

Let say, How many stuffed animals = N

3/4 of N stuffed animals = Number of button eyed ones.

But, PAM says the value of N is 15 which cannot be validated except Ashley

SQRT(49) = 7

1/288 =0.003472222222 in base 10

0.003472222222222222 x12 =0.04166666 =0.0 in base 12

0.04166666 x 12 =0.5 in base 12=0.00 in base 12

0.5 x 12 =6 =0.006 base12. So, we have:

1/288 =0.006 in base 12 and it terminates.

8(3n+6)-20n-40 how to solve

\(8\times(3n+6)-20n-40\)

\(=24n+48-20n-40=4n-8={\color{blue}4\times(n-2})\)  !

What is pi times 10 times 180/360 plus 10?

\(\pi\times10\times\frac{180}{360}+10 =5\pi+10 = {\color{blue}25.70796}\) !

3.6 times 1.1 over 1.2

\(3.6\times\frac{1.1}{1.2}\) = \(3\times1.1\)  = \({\color{blue}3.3}\)  !

THE SKY DURRR

You can probably do the numerator divided by the denominator 

How do you divide 399 divided by three?

\({\color{blue}\frac{399}{3}=133}\)  !

hi, so i'm trying to simplify this problem: x^2+5x+6 x x^2+11x +28 and I was hoping someone could give me a hand.

\({\color{red}x^2+5x+6x^2+11x+28}\)

= \(7x^2+16x+28\) = 0

   a            b           c

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(x = {-16 \pm \sqrt{256-4*7*28} \over 2*7}\)

\({\color{blue}x = \frac{-16\pm\sqrt {256-784}}{14}}\)

There is no real solution. 

Simplify:        [ (x² + 5x + 6) / (x² - 16) ] · [ (x² + 11x + 28) / ( x² + 2x - 3) ]

Factor:          [ (x + 3)(x + 2) ] / [ (x + 4)(x - 4) ] · [ (x + 7)(x + 4) ] / [ (x + 3)(x - 1) ]

Rearrange:  [ (x + 3)(x + 2)(x + 7)(x + 4) ] / [ (x + 4)(x - 4)(x + 3)(x - 1) ]

Cancelling the common terms:  [ (x + 2)(x + 7) ] / [ (x - 4)(x - 1) ]

If you wish, you can multiply out these factors.

Each face of a cube is a square.

If each edge of the square is  6.7  cm long, the area of each face is  6.7²  =  44.89 sq cm

Since there are  6  faces, the total surface area is  6 · 44.89  =  269.34 sq cm.

Solve:                                          | x/6 | < 1

Write as:                                -1  <  x/6  <  1

Multiply all three parts by 6:  -6  <  x  <  6

Of  62  students,  32  are freshmen,  32  are economic majors, and  12  are neither.

Subtract  12  from  62, giving  50  students who are either freshmen, economic majors or both.

If being a freshman is independent from being an economic major, adding the number of fresmen to the number of economic majors could, at most, be  50  students.

However, when you add  32  freshmen to the  32  economic majors, you get a total of  64 students. This means that   14  students must be both freshmen and economic majors (you get  14 by subtracting  50  from  64).

The probability that a student, drawn at random, is both an economics major and a freshman is:

14 /  62  =  7/31.

what is the basic number of pi

(-3)⁶ / (-3)⁸  =  (-3)^{6 - 8}  =  (-3)^-2  =  1 / (-3)²  =  1 / 9  =  1/9

If you divide number with the same base, subtract the exponents.

133

1/8 divided by 2=  1/16 b/c 1/8 divided by 2/1 would cause 2/1 to flip the fraction = 1/2. So 1/8 x 1/2 which is 1/16

25.71 (rounded) 

4/3

2*34 = 68 3*42 = 126