evaluate to the nearest hundreth 4^0.6? without using calculator.?

Answer 1

This is possible - you have to know negative binomial theorem.

Remember C(n,r) as the combinatorial "choose function" defined in the general case (not just positive integer n, r).

4^.6 = 2^1.2 = 2*2^.2.

By NBT,
2^.2 = (1+1)^.2 = C(.2,0)*(1^.2)*(1^0) + C(.2,1)*(1^-.8)*(1^1) + C(.2,2)*(1^-1.8)*(1^2) + ... = C(.2,0) + C(.2,1) + C(.2,2) + ...

Note:
C(.2,0) = 1
C(.2,1) = .2/1 = .2
C(.2,2) = (.2*-.8)/(1*2) = -.08
C(.2,3) = (.2*-.8*-1.8)/(1*2*3) = .048
C(.2,4) = (.2*-.8*-1.8*-2.8)/(1*2*3*4) = -.0336

each subsequent term is about -.8 times the preceding one, so the sum of all those terms is (by geometric series)
C(.2,0) + C(.2,1) + C(.2,2) + C(.2,3) + C(.2,4)*(1/1.8)
= 1 + .2 -.08 + .048 -.0186666
= 1.1493333

So 4^.6 rounded to 2 places is twice that number, or 2.30

This is a common type of high school math contest trick problem. You can use the same technique to approximate the n-th root of any small number by hand.

Click Best Answer if helpful. Thanks!

Answer 2

4 ^ 0.6
= 4 ^ (3/5)
= fifth-root-of( 4^3 )

Now, 4^3
(BE VERY CAREFUL ABOUT HOW YOU PRONOUNCE THAT! :-)
is 64.

So, what is the fifth root of 64?

Well, 64 is 2x2x2x2x2x2 or 2^6

Consider taking the 5th root of each 2,
then multiplying them back together.
Or just multiply the first five of them, to get 2.

In other words, the answer is the same as
two times the fifth root of 2.

Now, the square root of 2 is 1.4141
The square root of that is a bit less than 1.2
So, the 4th root of 2 is about 1.2,
and the 5th root is probably about 1.15

Doubling that gets about 2.3

That's not good enuf for the nearest 100th,
but perhaps you can refine it a bit further.

Good luck.!

.

Answer 3

yup mrobataille is correct. I don't wanna bore ya wit details

Answer 4

= 4^0.6
= 2.3

Answer: 2.3

Answer 5

2.297

haha...i didn't use my calculator...