Maintaining Mathematical Proficiency
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The product of two negative numbers is positive, but the product of a negative number and a positive number is negative.
See solution.
We are asked to consider two different expressions, - 4^n and (- 4)^n. We will analyze them one at a time for different values of n to see when the result is positive and when it is negative.
Since in this case the base is negative, it is convenient to review the result of a product with a negative factor.
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With this in mind, let's see what we get for some values of n.
| n value | Power | Sign |
|---|---|---|
| 0 | (-4)^0=1 | + |
| 1 | (-4)^1=- 4 | - |
| 2 | (- 4)^2=(- 4)(- 4)= 16 | + |
| 3 | (- 4)^3=(- 4)(- 4)(- 4)=- 64 | - |
| 4 | (- 4)^4=(- 4)(- 4)(- 4)(- 4)= 256 | + |
Can you identify any patterns? If n is even, we have pairs of negative factors which give a positive result. On the other hand, when n is odd, we have pairs of negative factors and a remaining negative factor. Consequently, for odd n-values the result is negative. We can summarize this as shown below.
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If the base of a power is negative, the result is positive if the exponent is even, otherwise the result is negative. |
Note that this is also true for negative integers, since we can always rewrite the power as a quotient with a positive exponent. (-4)^(- n) = 1/(-4)^n In these cases, the sign of the power will be given by the sign of the denominator, which follows the same rules we concluded above.
This case is different from the previous one, since here we are subtracting the result of the power 4^n. Then, as the base is positive, 4^n will always be positive, and therefore, - 4^n will always be negative.