Simplifying Positive Indices - Hey Folks!

[turnstile]

Hey folks.
I thought i might check my answers and confirm if theyre right or not.
the following have to be simplified and expressed with positive indices.
Below are the indices;

1. (√a^2b^3)^6

2. (x^a y^-b)^3 (x^3 y^2)^-a

3. (27x^3/8a^-3)^-2/3

4. {4√(x^-2/3 y^1/2)^3)}^-2/3

5. (4a^-2/ 9x^2)^1/2

6. (x * n√x 1/2)n^2/1-n

Any help would be greatly appreciated.
Thanks

 

[HallsofIvy]

You do understand, don't you, that it is impossible for us to check your answers and confirm if they are right or not when you didn't tell us what your answers were?

(There might be some evil people who would suspect that you don't really have any answers yourself but were hoping we would be foolish enough to post our own answers here so you could copy them. I would never think that myself! I mean it would be foolish to do that- someone might post wrong answers!)

 

[quicknote]

Hi there! It looks like you're trying to simplify expressions using positive indices. I'd be happy to check your answers for you.

1. (√a^2b^3)^6 = (a^2b^3)^3 = a^6b^9

2. (x^a y^-b)^3 (x^3 y^2)^-a = (x^3)^a (y^-b)^3 (x^3)^-a (y^2)^-a = x^3a y^-3b x^-3a y^-2a = x^3a-b y^-5a = x^-2a y^-5a

3. (27x^3/8a^-3)^-2/3 = (3^3 x^3 / 2^3a^-3)^-2/3 = (3x/2a)^-2/3 = (2a/3x)^2/3 = (2a)^2/(3x)^2 = 4a^2/9x^2

4. {4√(x^-2/3 y^1/2)^3)}^-2/3 = (4(x^-2/3 y^1/2)^3)^-2/3 = (4x^-2y^3/2)^-2/3 = (4y^3/2)^-2/3 (x^-2)^-2/3 = (4y^-1)^-2/3 x^4/3 = (4y)^2/3 x^4/3 = 4y^2/3 x^4/3

5. (4a^-2/ 9x^2)^1/2 = (4a^-2/9x^2)^1/2 = 4^1/2 (a^-2)^1/2 (9x^2)^-1/2 = 2 (a^-1)^1/2 (3x)^-1 = 2(a^-1/2) (3x)^-1 = 2/(a^1/2 * 3x)

6. (x * n√x 1/2)n^2/1-n = (x * x^(1/2))^(n^2/1-n) = x^(n^2/1