Power Laws and Indices

Question

Ok I have an exam and these stuff will be on it but i don't know how to work it out. So I would like you to give me the answer and the working out please. 

1. 8^2/3=
2. 24^-4/3=
3. (1/81)^-1/2=
4. (4/25)^-3/2=
5. (1/16)^-3/2=
6. (15 5/8)^4/3=

[  6.  (      5  )  ^4/3   

        (15 ___)

        (      8  )

(i dont know how else to present it sorry, hope it makes sense)  ]

 

 

Answer 1

This assignment is going to use a couple properties of exponents:

 

a^b/c = _^c√(a^b) = (_^c√(a))^b

a^-b = 1/(a^b)    OR    (a/b)^-c = (b/a)^c = b^c/a^c

 

 

 

1. 8^2/3 = (_³√(8))² = 2² = 4

 

2. 24^-4/3 = 1/(24^4/3) = 1/(_³√(24⁴)) = 1/(_³√(331776)) ≈ 1/69.228 ≈ 0.014445

 

3. (1/81)^-1/2 = 81^1/2 = √(81) = 9

 

4. (4/25)^-3/2 = (25/4)^3/2 = (√(25/4))³ = (5/2)³ = 125/8

 

5. (1/16)^-3/2 = 16^3/2 = (√(16))³ = 4³ = 64

 

6. (15 5/8)^4/3 = (125/8)^4/3 = (_³√(125/8))⁴ = (5/2)⁴ = 625/16