Identify the Exponent Law -
(a^2)^3 = a^2·3 = a^6

I think it's power of a power but all of the powers throw me off.

Question

Identify the Exponent Law - 
(a^2)^3 = a^2·3 = a^6

I think it's power of a power but all of the powers throw me off.

zehanz · Answer

Power of power law: $$(a^b)^c = a^{bc}$$Why does this hold?$$(a^b)^c=a^b \cdot a^b \cdot a^b \cdot ... \cdot a^b=a^{b+b+b+...+b}=a^{b \cdot c}$$So it makes use of another power law...

anonymous · Answer

Could it be product of powers?

zehanz · Answer

To clarify a bit more: in my explanation above, there are c factors a^b.
I then used the rule $$a^b \cdot a^c=a^{b+c}$$only with c times the same power, so you end up with c times an exponent b.

zehanz · Answer

So yes, it is power of powers, but that rule uses product of powers itself ;)

zehanz · Answer

Put differently: the power of powers rule is a consequence of the product of powers rule, which is a consequence of the definition of powers...

anonymous · Answer

Still a bit confusing,  but I got it! Thank you :)

zehanz · Answer

Yes, I know, it sounds confusing, but if you look into it, you'll see that it is quite simple, actually! (It's all about counting the factors of a power).