For what natural n>1 does the inequality 2^n>=n^3

Answer 1

2^n = n^3 with n = a natural number

2^10 = 2^3 x 2^3 x 2^3 x 2 = 8^3 x 2 = 64x16 = 1024

10^3 = 1000 that is close to 1024

2^9 = 512, 10^2 = 100 not close

not sure any natural number works for equality

but problem includes less than or equal to

so

n=1 doesn't work for 2^n>/=n^3 as 2 is less than 8

2^2 < 2^3 4 is not greater than 8

3^2 < 3^3

2^4 < 4^3

2^5 <5^3

2^6< 6^3

2^7<7^3

2^8<8^3

2^9<9^3 524 <729

2^10>10^3 1024>1000

2^11?11^3 2048>1331

probably all n =10 or greater

Answer 2

Try graphing each of these functions to see where the graph for 2^n >= the graph for n^3. Then remember that the question is limiting n to natural numbers > 1, i.e., n must be a counting number 2, 3, 4, ...

You may also make a table of values:

n: 2^n n^3

2 4 8

3 8 27

4 16 64

5 32 125

6 64 216

7 128 343

8 256 412 Continue until 2^n >= n^3

9