Each of the identities involving exponentiation has a corresponding identity involving multiplication. As exponentiation is repeated multiplication and multiplication is repeated addition, so exponential identities that also involve multiplication generally correspond to multiplicative identities that also involve addition. However, this rule doesn't apply to operations in the exponent (which is not multiplied but instead counts the multiplications); these operations are the same as in the multiplicative identity. Below I list the exponential identities together with their corresponding multiplicative identities. (Please keep in mind that I point out this correspondence only because it might help you to remember the new identities. If it's more confusing than it's worth, then forget about it; just learn the new identities for themselves.)

Exponential identity | Multiplicative identity |
---|---|

a^{0} ≡ 1; |
0a ≡ 0; |

a^{m}a^{n} ≡
a^{m+n} for
a ≠ 0 or m, n ≥ 0; |
ab + ac ≡
a(b + c); |

a^{1} ≡ a; |
1a ≡ a; |

(a^{m})^{n} ≡
a^{mn} for
a ≠ 0 or m ≥ 0; |
(ab)c ≡
a(bc); |

1^{n} ≡ 1; |
0a ≡ 0; |

(ab)^{n} ≡
a^{n}b^{n}; |
(a + b)c ≡
ac + bc; |

a^{−1} ≡ 1/a; |
(−1)a ≡ −a. |

For example, suppose that you want to simplify
*x*^{3} · *x*^{4}.
You can do this the long way,
using the definition of exponentiation as repeated multiplication,
and it's probably a good idea to try this a few times
until you get used to the exponential identities.
But you can also do this the quick way using the second identity above:

*x*^{3}·*x*^{4}≡ (*x*·*x*·*x*) (*x*·*x*·*x*·*x*) ≡*x*·*x*·*x*·*x*·*x*·*x*·*x*≡*x*^{7};*x*^{3}·*x*^{4}≡*x*^{3+4}≡*x*^{7}.

Similarly, suppose that you have (*x*^{3})^{4}.
Again, you can do this the long way or the short way:

- (
*x*^{3})^{4}≡ (*x*·*x*·*x*) (*x*·*x*·*x*) (*x*·*x*·*x*) (*x*·*x*·*x*) ≡*x*·*x*·*x*·*x*·*x*·*x*·*x*·*x*·*x*·*x*·*x*·*x*≡*x*^{12}; - (
*x*^{3})^{4}≡*x*^{3·4}≡*x*^{12}.

Now consider (*x**y*)^{4}:

- (
*x**y*)^{4}≡ (*x**y*) (*x**y*) (*x**y*) (*x**y*) ≡ (*x*·*x*·*x*·*x*) (*y*·*y*·*y*·*y*) ≡*x*^{4}*y*^{4}; - (
*x**y*)^{4}≡*x*^{4}*y*^{4}.

There are a few tricky situations where an exponent doesn't seem to appear.
In this case, use the identity *a* ≡ *a*^{1}.
For example, to simplify *x* · *x*^{4}:

*x*·*x*^{4}≡*x*(*x*·*x*·*x*·*x*) ≡*x*·*x*·*x*·*x*·*x*≡*x*^{5};*x*·*x*^{4}≡*x*^{1}·*x*^{4}≡*x*^{1+4}≡*x*^{5}.

An extreme version of this is to simplify *x* · *x*:

*x*·*x*≡*x*^{2};*x*·*x*≡*x*^{1}·*x*^{1}≡*x*^{1+1}≡*x*^{2}.

For example,
consider the expression *x*^{3}/*x*^{4},
that is *x*^{3} divided by *x*^{4}.
Since division is multiplication by a reciprocal
and (using the last identity above)
the reciprocal of a number
is the same as that number raised to the power of −1,
I can rewrite this
as *x*^{3}(*x*^{4})^{−1},
which I can then simplify using the other identities.
Thus, I get:

*x*^{3}/*x*^{4}— original expression;*x*^{3}(*x*^{4})^{−1}— rewritten without division;*x*^{3}*x*^{−4}— since −1 · 4 = −4;*x*^{−1}— since 3 − 4 = −1;- 1/
*x*— rewriting without negative exponents.

There is one tricky bit to watch out for in a problem like this one.
The two most commonly used exponential identities
(both of them used here:
the one where you multiply the exponents
and the one where you add the exponents)
have conditions on their usage.
When working with monomials, these conditions are always satisfied,
since the exponents in a monomial are always whole numbers.
Here, however, the conditions fail unless *x* ≠ 0.
So I need to check and make sure that all the expressions above
are really equivalent even when *x* := 0.
In this case, they are;
all of the expressions are undefined, one way or another,
when *x* := 0.
So there's nothing to worry about in this case.

Now consider *x*^{4}/*x*^{3}.
This is very similar to the previous example,
but now I will need to worry about the possibility that *x* := 0.
Watch as I go through almost the same reasoning:

*x*^{4}/*x*^{3}— original expression;*x*^{4}(*x*^{3})^{−1}— rewritten without division;*x*^{4}*x*^{−3}— since −1 · 3 = −3;*x*^{1}for*x*≠ 0 — since 4 − 3 = 1, and*x*^{−3}is undefined when*x*:= 0;*x*for*x*≠ 0 — since raising to the power of 1 has no effect.

Here's a more complicated example showing in more detail how to keep track of these things:

*x*^{2}*y*^{4}/*x*^{4}*y*^{2}— original expression (for this new example);*x*^{2}*y*^{4}*x*^{−4}*y*^{−2}— rewritten without division;*x*^{2}*x*^{−4}*y*^{4}*y*^{−2}— rearranging multiplication;*x*^{−2}*y*^{2}for*y*≠ 0 — since 2 − 4 = −2, 4 − 2 = 2, and*y*^{−4}is undefined when*y*:= 0;*y*^{2}/*x*^{2}for*y*≠ 0 — rewritten without negative exponents.

Finally, I should tell you how all of the examples in this section
can be easily done in a single step.
The basic idea is this:
*Division of a power
corresponds to multiplication with the opposite exponent.*
Since multiplying powers involves adding the exponents,
this means that dividing powers involves *subtracting* the exponents.
So in the first example, *x*^{3}/*x*^{4},
you simply calculuate 3 − 4 = −1
to get the answer *x*^{−1},
which you can rewrite as 1/*x* if you wish.
In the next example, *x*^{4}/*x*^{3},
you calculate 4 − 3 = 1;
noticing that the result should be undefined when *x* := 0,
your answer is *x* for *x* ≠ 0.
Finally, in the example
*x*^{2}*y*^{4}/*x*^{4}*y*^{2},
you calculate 2 − 4 = −2 for the exponent on *x*,
4 − 2 = 2 for the exponent on *y*,
and notice that the result should be undefined
even when *y* := 0,
to get the result
*x*^{−2}*y*^{2} for *y* ≠ 0,
which you can rewrite
as *y*^{2}/*x*^{2} for *y* ≠ 0
if you wish.

For example, consider the monomial
3[2(*x**y*)^{4}*x*]^{3}*y*^{2}.
This is pretty busy, but I can handle it step by step:

- 3[2(
*x**y*)^{4}*x*]^{3}*y*^{2}— original expression; - 3[2
*x*^{4}*y*^{4}*x*]^{3}*y*^{2}— distribute the exponent 4 to the factors*x*and*y*; - 3[2
*x*^{5}*y*^{4}]^{3}*y*^{2}— combine the factors*x*^{4}and*x*into*x*^{5}(remember that*x*≡*x*^{1}); - 3[2]
^{3}[*x*^{5}]^{3}[*y*^{4}]^{3}*y*^{2}— distribute the exponent 3 to the factors 2,*x*^{5}, and*y*^{4}; - 3[8]
*x*^{15}*y*^{12}*y*^{2}— work out 2^{3}and multiply exponents to simplify powers of powers; - 24
*x*^{15}*y*^{14}— combine the constant factors and the*y*factors.

You can apply these same ideas
to examples with division or negative exponents,
as long as you watch out for division by zero
(or raising zero to a negative exonent).
For example, consider the monomial above,
now with negative exponents instead of positive ones:
3[2(*x**y*)^{−4}*x*]^{−3}*y*^{−2}.
Because of the negative exponents, this expression is *not* a monomial,
but I can simplify it anyway:

- 3[2(
*x**y*)^{−4}*x*]^{−3}*y*^{−2}— original expression; - 3[2
*x*^{−4}*y*^{−4}*x*]^{−3}*y*^{−2}— distribute the exponent −4 to the factors*x*and*y*; - 3[2
*x*^{−3}*y*^{−4}]^{−3}*y*^{−2}— combine the factors*x*^{−4}and*x*into*x*^{−3}(remember that −4 + 1 = −3); - 3[2]
^{−3}[*x*^{−3}]^{−3}[*y*^{−4}]^{−3}*y*^{−2}— distribute the exponent −3 to the factors 2,*x*^{−3}, and*y*^{−4}; - 3[1/8]
*x*^{9}*y*^{12}*y*^{−2}for*x*≠ 0 — work out 2^{−3}and multiply exponents to simplify powers of powers; - (3/8)
*x*^{9}*y*^{10}for*x*,*y*≠ 0 — combine the constant factors and the*y*factors.

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