Negative Bases in Sums

Powers with negative bases can sit inside longer lines. Each power is worked out first, with the even or odd rule deciding its sign, and then the line is an ordinary sum of integers. Take (-4)^3 + 5^2.

(-4)^3 has an odd exponent, so it is negative, and its size is 4^3, which is 64: so (-4)^3 is -64. 5^2 is 25. The line becomes -64 + 25, and adding numbers with different signs gives -39.

Deciding the sign first and the size second keeps the work small. You do not have to multiply a chain of negatives one factor at a time: count the exponent for the sign, then work out the power of the plain number for the size.