Exponent Rule Question: 12^5 = (2^10)(3^5)

[katejohn7]

For some reason I am having difficulty understanding why 12^5 = (2^10)(3^5) - how do you get there? I can't seem to break it down..

Thank you for your help. I did not see this question posted. This is from page 68 of Number Properties.

[georgepa]

(12)^5
= (4 . 3 )^5
= (4^5) . 3^5 => Since a^x . b^x = (ab)^x
= (2 . 2)^5 . 3 ^5 => Reduce 4 to 2 . 2
= (2^5 . 2^5) . [3^5 ] => Since a^x . b^x = (ab)^x
= 2^(5+5) . 3^5 => Since x^a . x^b = x^(a+b)
= 2^(10) . 3^5

[Ben Ku]

Thanks georgepa breaking down this expression. The key to being efficient with exponent questions is to be (1) Break things down to prime factors, and (2) Be very fluent with the laws of exponents. If you practice these two, you should do find on similar problems.