Confused by exponents when trying to simplify within a square root

[rstratton11]

I’m confused as to why for a^6 that it would break down as (a^3)^2, because isn’t 3^2 = 9? Why are we multiplying the exponents?

 

[skeeter]

[imath](a^3)^2 = a^3 \cdot a^3[/imath]

 

[Steven G]

Yes, 3^2 = 9.
In 3^2 we are squaring 3.
In (a^3)^2 we are not squaring 3, rather we are squaring a^3. (a^3)^2 = a^3*a^3 = a^6

(a^m)^n = (a_1*a*_2*...*a_m)_1 *(a_1*a*_2*...*a_m)_2*...*(a_1*a*_2*...*a_m)_n = a^(mn)

 

[rstratton11]

Yes, 3^2 = 9.
In 3^2 we are squaring 3.
In (a^3)^2 we are not squaring 3, rather we are squaring a^3. (a^3)^2 = a^3*a^3 = a^6

(a^m)^n = (a_1*a*_2*...*a_m)_1 *(a_1*a*_2*...*a_m)_2*...*(a_1*a*_2*...*a_m)_n = a^(mn)

Ok, that’s making much more sense to me than before. Thank you!