Question

How do you write `sqrt(24a^10b^6)` in simplest form?

Answer 1

You can write it as:
`sqrt(6*4*a^(10)b^6)=2a^5b^3sqrt(6)`

Where you used the fact that `sqrt(x)=x^(1/2)`

and so:
`sqrt(6*4*a^(10)b^6)=(2^2*6*a^(10)b^6)^(1/2)=`
`=2^(2*1/2)6^(1/2)a^(10*1/2)b^(6*1/2)=2a^5b^3sqrt(6)`

Answer 2

If we may assume that `a` and `b` are non-negative, then we write `sqrt(24a^10b^6)=2a^5b^3sqrt6`

`sqrt(24a^10b^6)=sqrt(4*6*(a^5)^2(b^3)^2)=2sqrt6a^5a^3=2a^5b^3sqrt6`

The key is the repeated use of: `sqrt(n^2)=n`
(for non-negative `n`).

So `sqrt4=sqrt(2^2)=2`, and `sqrt((a^5)^2)=a^5` and `sqrt((b^3)^2)=b^3`.

(If we may NOT assume `a` and `b` are non-negative, the we need absolute values, because then we would need `sqrt(a^2)=absa`
`sqrt(24a^10b^6)=2abs(a^5b^3)sqrt6`.)

Answer 3

Extract the largest square from each of the terms within the square root:
`24 = 2^2 * 6`
`a^(10) = (a^5)^2 *(1)`
`b^6 = (b^3)^2 *(1)`

So
`sqrt(24 a^(10)b^6)`

`=sqrt( (color(red)(2)^2)(6)color(red)((a^5))^2color(red)((b^3))^2)`

`= color(red)(2a^5b^3) sqrt(6)`