What are the powers of 4 through 1000

{4^0, 4^1, 4^2, 4^3, 4^4}, or, with a bit of rounding off,

{4^0, 4^1, 4^2, 4^3, 4^4, 4^5} = {1, 4, 16, 64, 256, 1024}

Step-by-step explanation:

I wanted to determine the largest power of 4 that does not exceed 1000.  Here's what I did:  

4^x = 1000.  Taking the ln of both sides, we get x ln 4 = ln 1000.

                               6.9078

Solving for x:  x = = 4.98

                                1.3863

Thus, the answer to this question is {4^0, 4^1, 4^2, 4^3, 4^4}.  If you're willing to talk in terms of approximations, then there'd be one more power of 4 in this set:  4^5.  Note that 4^5 = 1024, whidh is just a bit over 1000.

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation:

5   1/4   miles per hour

step-by-step explanation:

15   3/4 / 3 = 5   1/4