Now you can move on to exponents, using the cancellation-of-minus-signs property of multiplication.
Recall that powers create repeated multiplication. For instance, (3)^{2} = (3)(3) = 9. So we can use some of what we've learned already about multiplication with negatives (in particular, we we've learned about cancelling off pairs of minus signs) when we find negative numbers inside exponents.
For instance:
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The square means "multiplied against itself, with two copies of the base". This means that I'll have two "minus" signs, which I can cancel:
(–3)^{2} = (–3)(–3) = (+3)(+3) = 9
Pay careful attention and note the difference between the above exercise and the following:
–3^{2} = –(3)(3) = –1(3)(3) = (–1)(9) = –9
In the second exercise, the square (the "to the power 2") was only on the 3; it was not on the minus sign. Those parentheses in the first exercise make all the difference in the world! Be careful with them, especially when you are entering expressions into software. Different software may treat the same expression very differently, as one researcher has demonstrated very thoroughly.
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(–3)^{3} = (–3)(–3)(–3)
= (+3)(+3)(–3)
= (9)(–3)
= –27
(–3)^{4} = (–3)(–3)(–3)(–3)
= (+3)(+3)(–3)(–3)
= (+3)(+3)(+3)(+3)
= (9)(9)
= 81
(–3)^{5} = (–3)(–3)(–3)(–3)(–3)
= (+3)(+3)(–3)(–3)(–3)
= (+3)(+3)(+3)(+3)(–3)
= (9)(9)(–3)
= –243
Note the pattern: A negative number taken to an even power gives a positive result (because the pairs of negatives cancel), and a negative number taken to an odd power gives a negative result (because, after cancelling, there will be one minus sign left over). So if they give you an exercise containing something slightly ridiculous like (–1)^{1001}, you know that the answer will either be +1 or –1, and, since 1001 is odd, then the answer must be –1.
You can also do negatives inside roots and radicals, but only if you're careful. You can simplify , because there is a number that squares to 16. That is,
...because 4^{2} = 16. But what about ? Can you square anything and have it come up negative? No! So you cannot take the square root (or the fourth root, or the sixth root, or the eighth root, or any other even root) of a negative number. On the other hand, you can do cube roots of negative numbers. For instance:
...because (–2)^{3} = –8. For the same reason, you can take any odd root (third root, fifth root, seventh root, etc.) of a negative number.
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