Exponents with Negative Base Values:


When multiplying negative numbers together it is necessary to use parentheses,
such as (-4)(-4)(-4)(-4)(-4)(-4) to avoid confusion.

When raising a negative number to some power it is necessary to use parentheses,
such as (-4)6 if you want the negative value raised to the power.

confused2 Here's the problem about the parentheses:
(-4)6 and -46 are not the same.
We know (-4)6 = (-4)(-4)(-4)(-4)(-4)(-4) = 4096.

But, -46 = - (4)(4)(4)(4)(4)(4) = -4096

The problem is that since there are no parentheses in -46, it is being viewed as (-1) times 46. Order of operations dictates that the exponent be done before the multiplication. You can think of it as the exponent 6 being STUCK to the 4, but not to the negative sign.
In (-4)6, the exponent 6 is STUCK to the parentheses, and since the parentheses are done first, the 4 gets negated before the exponent is applied.

More Information: The problem we just encountered appeared because the exponent was an even power. If the exponent were an odd power, we could arrive at a correct answer. But, as shown below, -43 is raising 4 to the third power, and then negating the result. It is not truly raising (-4) to the third power, like (-4)3 is doing.

(-4)3 = (-4)(-4)(-4) = -64
-43 = -(4)(4)(4) = -64

Since it is usually the intent in these situations to "raise the negative base to the exponent power", avoid confusion and use parentheses!!

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More about exponents being EVEN or ODD:

Even powers of negative numbers allow for the negative values to be arranged in pairs. This pairing guarantees that the answer will always be positive. Remember, a negative number times another negative number yields a positive result.

(-3)6 = (-3)(-3) (-3)(-3) • (-3)(-3)
= 9 • 9 • 9
= 729

Odd powers of negative numbers, however, always leave one factor of the negative number not paired. This one lone negative term guarantees that the answer will always be negative.

(-3)5 = (-3)(-3) • (-3)(-3) • (-3) ← lone factor
= 9 • 9 • (-3)
= -81

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So, what have we discovered about using parentheses?
If a negative base is enclosed in parentheses:
  • the result is positive if the exponent is even.
  • the result is negative if the exponent is odd.

If a negative base is not enclosed in parentheses:
  • the result is always negative.

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For calculator help with exponents
click here.


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