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Question

a^3–a

a³–a

Answer 1

a^3–a
a³–a

 

These two expressions both represent:    a*a*a - a

  They are written differently because keyboards and computer languages have differences.  The carrot ("^") was a handy in-line way to write a superscript when keyboards did not allow superscripts or subscripts.  You might also find "**" for exponentiation, and SQRT(x), and other such notation.

 

You use the distributive property to factor such an expression:

    a*(b+c) = a*b + a*c             [note: also good for subtraction]

 

But, use it in reverse:

    a*b + a*c = a*(b+c)

 

This problem is simply:

    a*a*a - a

    a*a*a - a*1

    a*(a*a - 1)

    a(a²-1)

 

Now, if you want to factor completely, you might recognize that (a²-1) is the difference of squares and factors to (a+1)(a-1), so the final result is:

    a(a+1)(a-1)          

Answer 2

a^3–a = a(a^2–1) = a(a+1)(a-1)