Find all the powers of 3 that are in the range from 3 to 1,000.

A power is how many times a number is multiplied by itself. So for 3, the number given, the values are as follows: 3x3=9 3x3x3=27 3x3x3x3=81 3x3x3x3x3=243 3x3x3x3x3x3=729 The next power will be greater than 1000 and therefore out of the range given.

Answered by Kalahira | 2024-06-24

Smallest power: 3 1 = 3 .[/tex] Largest: 3^6 = 729\ Powers within range: 3, 9, 27, 81, 243, 729\
here's a more detailed step-by-step approach:

Identify the smallest power of 3 within the range:

- The smallest power of 3 greater than or equal to 3 is 3^1 = 3\

Find the largest power of 3 within the range:

- We need to find the largest integer power of 3 that does not exceed 1000.
- We can use logarithms to solve for this. The largest power of 3 less than or equal to 1000 can be found by solving the inequality 3^n \leq 1000\ where [tex] n is the power.
- Taking logarithms to base 3 of both sides, we get:
lo g 3 ​ ( 3 n ) ≤ lo g 3 ​ ( 1000 ) n ≤ lo g 3 ​ ( 1000 )
- Now, calculate lo g 3 ​ ( 1000 ) using a calculator:
lo g 3 ​ ( 1000 ) ≈ lo g ( 3 ) lo g ( 1000 ) ​ ≈ 6.2877
- Since n must be an integer, the largest integer less than or equal to 6.2877 is 6. So, n = 6\
- Therefore, the largest power of 3 within the range is 3^6 = 729\

Generate all powers of 3 within the range:

- Now that we have both the smallest and largest powers of 3 within the range, we can list all the powers of 3 from 3^1\) to \(3^6\
- So, the powers of 3 within the range 3 through 1000 are:
3 1 = 3 3 2 = 9 3 3 = 27 3 4 = 81 3 5 = 243 3 6 = 729
These are all the powers of 3 within the range from 3 through 1000.

Answered by vinithaira | 2024-06-25

The powers of 3 from 3 to 1,000 are 3, 9, 27, 81, 243, and 729. The next power, which is 2187, exceeds this range. Hence, these six values are all the valid powers of 3 within the specified limits.
;

Answered by Kalahira | 2024-10-10