Find the cube root of 512 by prime factorisation

To do: Find the cube root of 512 by prime factorization.

Solution:

 Prime factors of $512 =2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

Prime factorization of $512  = 2^9$  

cube root of $512  =  (512)^{\frac{1}{3}}   =  ( 2^9  )^{\frac{1}{3}}$   

$(a^m  )^n  = a^{m\times n}$ 

$( 2^9  )^{\frac{1}{3}}  =  ( 2 )^{9\frac{1}{3}}$  

=$ 2^ 3$ 

= 8

Therefore the Cube root of 512 is 8