geno3141

12^3  =  12 x 12 x 12  =  1728

It also equals (3·4)^12  =  3^12 · 4^12

Plus a lot of other expressions; just what are you looking for?

It's still 19.

I'm picturing something that looks like a Rubik's cube, but each side is  4 x 4.

The top side is  4 x 4             (so 16 cubes); 

the bottom side is also 4 x 4  (so another 16 cubes)

the front side is  4 x 4  but  we already have cubes on the top and on the bottom,                                                                                       so all we need are another  8 cubes

same for the back side           (so another  8  cubes).

for the left side, we have its top row, its bottom row, its left column and its right column, 

                                            so all we need are  4  more cubes

same for the right side          (so another 4  cubes )

For a total of 56 cubes.

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Another way to look at this (perhaps easier) is to find the total number of cubes necessary if it were hollow:        4 x 4 x 4  =  64

And then subtract out the inside; which is a  2 x 2 x 2  cube  =  8 

For a total of  64 - 8  =  56  cubes

If  x  is the smallest odd integer, then  x + 2  will be the next larger odd integer, and  x + 4  will be the largest of the three odd integers.

4[ (x) + (x + 4) ] = 5 + 3(x + 2)

4( 2x + 4 )  =  5 + 3x + 6

8x + 16  =  3x + 11

5x  =  -5

So the smallest integer is  -1.

Use this to find the other two.

If we call the numbers  x  and  y:

                                               x + y  = -5

                                               x - y  =  13

Add down the columns to get:  2x      =  8,  which means that  x  =  4.  

Then, using the equation  x + y  =  -5     --->     4 + y  =  -5     --->     y  =  -9  

-2 - 37    is    -2 + -37    which equals  -39

To remove a  log  use the  (2nd) log  feature of your calculator or use  10^x.

if  log K = 18  then  k = 10^18 

Prime numbers are:  2, 3, 5, 7, 11, ...

Start with 2:  900 ÷ 2  =  450

                    450 ÷ 2  =  225

                    225 ÷ 2  =  a fraction; so, don't count this

Next, try 3:  225 ÷ 3  =  75

                     75 ÷ 3  =  25

                     25 ÷ 3  =  a fraction; so, don't count this

Next, try 5:    25 ÷ 5  =  5

                      5 ÷ 5  =  1  We're done!

The prime factors are 2, 3, and 5.

The prime fractorization is:  2² · 3² · 5²   (the prime factors with the number of times each one is used)

9 + 10  =  19.

"the product of four and a numer" means to multiply 4 times a number. Since we don't know what the number is, we'll use a variable, like 'n'. So, we now have:   4·n,  or just  4n.

We now need to subtract  12  from this; so we get:  4n - 12.