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HA!!!!!!!!!!!

Can I get a HOOOYA?

Another one for me to study.

Thanks Chris :)

Thanks Chris,  

I had not thought about that before.

To put it another way..

If the decimal is to terminate then

The prime factors of the denominator must ONLY  be the prime factors of 10 

 (which are  2 and 5 )

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

I assume this can be extended to other bases.

So

1/5 base 8 is a recurring decimal because the prime factors of 5 are not the prime factors 8

That works.

I just did the division

and got 

\(\frac{1}{5_8}=0.14\bar6\bar3\)   (base8)    

How about that!     

Thanks Chris       

Hiya!!! 👋

In a reduced form.....If we can express the denominator solely as a power of 2 or a power of 5 - or both - the fraction will terminate....else..it repeats.....

1/ 40  terminates  = 1 / [ 2^3 * 5]

1/ 24 does not =  1 / [ 2^3 * 3 ]

2/125 terminates  = 2 / 5^3

2/ 99 does not  =  2 / [ 3^2 * 11]  

7/ 300  does not =  7 / [ 2^2 * 5^2 * 3 ] 

7 / 200  terminates  =  7 / [ 2^3 * 5^2]

Here's the provision for the reduced form

3 / 300     appears to not terminate....however

3 / 300  =   1 / 100  =    1 /  [  2^2 * 5^2 ]    so this actually does terminate  !!!

And that's all there is to it  !!!!

Here's a fast way ....we multiply by the base and add the next digit.....etc.

Suppose we have

1011₂

Multiply the leading "1" by 2 and add the next digit  ⇒  2 * 1 + 0  = 2

Multiply this result by 2 and add the next digit ⇒   2 * 2 + 1  =  4 + 1  = 5

Multiply this result by 2  and add the final digit ⇒ 2 * 5 + 1  = 10 + 1  = 11  

Verify for yourself that  1011₂   =  11₁₀

BTW - This method will convert any base to a base 10 number......

8 ÷ (10−9) 3 × 7 + 4

There is a symbol missing between the bracket and the 3 maybe i3 is a power ?

8 ÷ (10−9)^3 × 7 + 4

\(8 ÷ (10−9) ^3 × 7 + 4\\ =8 ÷ (1) ^3 × 7 + 4\\ =8 ÷ 3× 7 + 4\\ =\frac{8 }{3}× 7 + 4\\ =\frac{8\times7 }{3}+ 4\\ =\frac{56}{3}+ 4\\ =18\frac{2}{3}+ 4\\ =22\frac{2}{3}\\\)

So i figured it out by looking at my number 1  kinda weird lol but thanks 

12((3x-4)/(3)-(3-4x)/(4)-(6-2x)/(6))

\(=\,12(\frac{3x-4}{3}-\frac{3-4x}{4}-\frac{6-2x}{6})\)

\(=\,\frac{(12)(3x-4)}{3}-\frac{(12)(3-4x)}{4}-\frac{(12)(6-2x)}{6} \\~\\ =\,\frac{(4)(3x-4)}{1}-\frac{(3)(3-4x)}{1}-\frac{(2)(6-2x)}{1} \\~\\ =\,(4)(3x-4)-(3)(3-4x)-(2)(6-2x) \\~\\ =\,12x-16 - 9+12x -12+4x \\~\\ =\,28x-37\)            Reduce the fractions.

Use the same formula that I used in the first question after 88 years.

Your Interest rate should be 4.4% APR and NOT 44% !!!!.

Yolanda's Account:

FV =$4,651,166.97 - Balance of Yolanda's account after 77 years.

Zach's Account:

FV =$4,342,644.09 - Balance of Zach's account after 77 years. 

This is the formula you would use to compare the two:

Your interest rate should be 5.5% APR and NOT 55% APR !!!.

FV = P x {[1 + R]^N - 1 / R}

Yolanda's Account:

FV = 100 x {[1 + 0.055/12]^(88*12) - 1 / (0.055/12)}

FV = 100 x {[1.00458333....]^1,056 - 1 / (0.00458333....)}

FV = 100 x { 124.0785711 / 0.00458333}

FV = 100 x                      27,071.6882353.......

FV =$2,707,168.82 - The balance of Yolanda's account after 88 years.

Zach" Account:

FV =1,000 x {[1 + 0.055]^88 -1 / 0.055}

FV =1,000 x {[1.055]^88 - 1 / 0.055}

FV =1,000 x          2,004.15625580...........

FV =$   2,004,156.26 - The balance of Zach's account after 88 years.    

ax^2 + bx + c     (1)

In this form, the x coordinate of the vertex is given by

 - b / [ 2* a ]         

To find the y coordinate, sub this result back into   (1)  

Examlple

2x^2 +  8x +  `10     ......        a = 2       b =  8

x coordinate of the vertex  =  -8/ [2 * 2 ]  =  -8 / [ 2 * 2 ]  =  -8 / 4  =  -2

y coordinate of the vertex =  2 (-2)^2 + 8(-2) + 10  =      2

THANK YOU!!!

(t-1)^3 + 6(t-1)^2 + 12(t-1) + 8

Let's write this in a different manner

(t-1)^3  + 8  + 6(t-1)^2 + 12(t-1) 

The first two terms can be written as a sum of cubes thusly

[ (t - 1) + 2 ]  [ (t - 1)^2  - 2(t - 1)  + 4 ]  =

[ t + 1 ] [ t^2 - 2t + 1 - 2t + 2 + 4 ]  =

[ t + 1]  [  t^2 - 4t + 7 ]      (1)

The second pair of terms can be factored as

6 [ t - 1 ] [ t - 1 + 2]  =

6 [ t - 1] [t + 1]  =  [ t + 1] [6t - 6]    (2)

Putting (1) and (2) together, we have

[ t + 1 ]  [  t^2 - 4t + 7 + 6t - 6 ]  =

[ t + 1 ]  [ t^2 + 2t + 1 ]  =

[  t + 1 ] [t + 1] [t + 1 ]  =

[ t + 1]^3

The cost of a telephone call is $0.75 + $0.25 times the number of minutes.

The cost of a telephone call   =   $0.75  +  $0.25  *  the number of minutes.

The number of minutes  =  t , so...

The cost of a telephone call  =  $0.75 + $0.25t

Or we can say

The cost of a telephone call, in dollars  =  0.75 + 0.25t        

-2  is less than  1

-2  <  1

    \(1\frac25-\frac48\)

                                  \(\frac48\)  is the same as  \(\frac12\)

=  \(1\frac25-\frac12\)

                                  \(1\frac25\)  is the same as   \(1+\frac25\)

=  \(1+\frac25-\frac12\)

                                  Now we need a common denominator. Multiply and divide  1  by  10 .

=  \(\frac{10}{10}+\frac25-\frac12\)

                                  Multiply and divide  \(\frac25\)  by  2 .

=  \(\frac{10}{10}+\frac{2*2}{5*2}-\frac12\)

=  \(\frac{10}{10}+\frac{4}{10}-\frac12\)

                                  Multiply and divide  \(\frac12\)  by  5 .

=  \(\frac{10}{10}+\frac{4}{10}-\frac{1*5}{2*5}\)

=  \(\frac{10}{10}+\frac{4}{10}-\frac{5}{10}\)

10 apples + 4 apples - 5 apples  =  9 apples   So

10 tenths + 4 tenths - 5 tenths  =  9 tenths

= \(\frac9{10}\)

What is the math problem?

They are indicating where the graphs cross a particular axis.....I have to admit, that threw me at first, too, ISG  !!!!......I don't even see why they are necessary in this problem !!!

F  is correct.....the graphs intersect in two places  [two x values ]

I can explain why the others are incorrect.....if you want me to.......!!!

I totally get what you are saying but what are the little numbers around it. Are they the coordinates or something?

The answer cannot be graphs   F , G, H  or  K

The reason is that some of the "shaded" portion  of each graph falls into the 3rd quadrant where x and y are both negative  which  would make   1  < x + y < 2     false

So...by default.....the answer is   J