Powers of Negative Numbers

A negative base works the same way. You count the factors, and you use the sign rules you already know. (-2)^2 is (-2) × (-2). Two negative factors multiply to a positive number, so the result is 4. (-2)^3 has one more negative factor, which flips the sign. The result is -8.

Each extra factor of a negative number flips the sign once, so the exponent alone decides the sign. An even exponent gives a positive result. An odd exponent gives a negative result. (-1)^100 is 1, and (-1)^101 is -1.

The size of the answer does not depend on the sign. (-2)^5 has the same size as 2^5, which is 32. Work out the size by counting factors. Then check whether the exponent is even or odd to choose the sign. The exponent 5 is odd, so (-2)^5 = -32.