Are (-a)^3 and -a^3 the Same Thing?

Date: 11/08/2005 at 22:32:27
From: Alex
Subject: test question in 7th grade

(-a) times (-a) times (-a) = ?

I answered (-a) to the third power.  My teacher said NO, the answer is
-a to the third power (same answer without the parantheses).

He said (-a) to the third power is NOT equal to -a to the third power.
Is he correct?  If so, why?

Date: 11/10/2005 at 23:48:35
From: Doctor Wilko
Subject: Is 'negative a' cubed equal to the negative of 'a cubed'? 

Hi Alex,

Thanks for writing to Dr. Math!

I'll give a short answer, and then I'll elaborate more below.

I'd say your answer of (-a)^3 IS INDEED a valid answer to the question 
as you posed it.  

But if your teacher wanted you to SIMPLIFY, then he may have been 
looking for -a^3 as the correct answer.  (I'll show why (-a)^3 
simplifies to -a^3 below!)

If you were just told that your answer was wrong without a good reason 
as to why, then I'd say a better response to your answer could have 
been, "Alex, you are right that [-a * -a * -a] DOES equal (-a)^3. BUT 
because of the odd exponent, (-a)^3 can be further simplified to 
-a^3."

Does the distinction above make sense?

==================

Allow me to elaborate on the question a little more.

The MEANING of the two expressions can be interpreted as different if 
the focus is on the form of the expressions and how the order of 
operations applies to exponents and multiplication.

  (-a)^3 means "To cube 'negative a'," whereas

  -a^3 means "To cube 'a', and then negate that."

However,

(-a)^3 and -a^3 will give the SAME ANSWER if you plug in a value for 
'a' (because of the odd power).  

For example, let a = 2,

  (-2)^3 = (-2)*(-2)*(-2) = -8, and

  -(2)^3 = -[2*2*2] = -[8] = -8

Does this always hold?  

With odd powers, the answer is yes.  See the algebraic argument below.

===================

Is

  (-a)^3  ['negative a' -- cubed]

equal to

  -(a)^3  [the negative of -- 'a cubed']?

---------------------

  (-a)^3 =              ('Negative a' -- cubed)

  -a * -a * -a =

  (-1 * a) * (-1 * a) * (-1 * a) = 

  -1 * -1 * -1 * a * a * a =

  -1 * a * a * a =

  -1 * a^3 = 

   -a^3                (The negative of -- 'a cubed')

This shows that the two expressions are equal.

==================

Note that this equality DOESN'T hold for even powers.

A similar algebraic argument will show that

 (-a)^2 is NOT the same as -a^2

See this link from our archives for more on this:

  Negative Squared, or Squared Negative?
    http://mathforum.org/library/drmath/view/61633.html 

===================

Does this help?  Please write back if you have further questions.
:-)


- Doctor Wilko, The Math Forum
  http://mathforum.org/dr.math/