Is −3² equal to −9 or to 9?

BIDMAS works indices before adding and subtracting, and the minus in front of 3^² counts as a subtraction, so it waits until the square is done.

Only a bracket puts the minus inside the squaring. The minus of a subtraction waits the same way: 5 − 3^² = 5 − 9 = −4.

The index says how many negatives are multiplied

In (−2)^⁵ the bracket puts the minus inside, so every factor the index counts is negative.

Two negatives multiplied together make a positive, so the factors pair off into positive products. An odd index leaves one with nothing to pair with, and that one turns the answer negative.

Common mistakes

• Squaring the minus along with the number −4^² is not 16. The index stands against the 4 alone, so the squaring gives 16 and the minus is still in front of it: −4^² = −16, while (−4)^² = 16.
• Keeping a minus because the base was negative (−3)^⁴ is not −81. Four negatives make two pairs, and each pair is positive, so nothing is left to carry a minus: the answer is 81.
• Taking every power of a negative number to be positive That holds for squares, because two negatives pair off. Three negatives leave one unpaired, so (−3)^³ = −27, not 27.

Summary

1. BIDMAS works the index before the minus, so −3^² = −9.
2. A bracket gathers the minus in with the number, so (−3)^² = 9.
3. The index counts the negative factors: an even index leaves the answer positive, an odd one leaves it negative.
4. The size is the same power of the number without its sign: 2^⁵ = 32.