Question

The expression (a3)2 can be simplified as shown here: (a3)2 = (a3)(a3) = (a · a · a)(a · a · a) = a6 Write out in words how you would explain each step of this process to a friend. Why is each step equal to the previous step? Using a similar reasoning, explain why (a2)3= a6.

Answer 1

1 step: if you have to square some number, then you have to multiply this number by itself:

$(a^3)^2=a^3\cdot a^3.$

2 step: to find the cube of a number you should multiply this number two times by itself:

$a^3=a\cdot a\cdot a.$

3 step: multiplying a by itself 5 times you get the 6th power of a:

$(a\cdot a\cdot a)\cdot (a\cdot a\cdot a)=a\cdot a\cdot a\cdot a\cdot a\cdot a=a^6.$

Similar way to count $(a^2)^3$ is:

$(a^2)^3=a^2\cdot a^2\cdot a^2=(a\cdot a)\cdot (a\cdot a)\cdot (a\cdot a)=a\cdot a\cdot a\cdot a\cdot a\cdot a=a^6.$