Now you can move on to
exponents,
using the cancelation-of-minus-signs property of multiplication. For instance,
(3)^{2}
= (3)(3) = 9. In the
same way:

Simplify (–3)^{2}

(–3)^{2}
= (–3)(–3) = (+3)(+3) = 9

Note the difference between
the above exercise and the following:

Simplify –3^{2}

–3^{2}
= –(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 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.

You can also do negatives
with roots,
but only if you're careful. You can do ,
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.