A quantity with an exponent has three components--the
base, the
exponent,
and the coefficient.

- In the quantity 3
*x*^{5}, the coefficient is 3, the base is *x*, and
the exponent is 5.

- In the quantity 3(16)
^{7x}, the coefficient is 3, the base is
16, and the exponent is 7*x*.

- In the quantity 26(2
*y*)^{xy}, the coefficient is 26, the base is 2*y*,
and the exponent is *xy*.

- In the quantity
*r*^{2}, the (implied) coefficient is 1, the base is
*r*, and the exponent is 2.

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Adding and Subtracting Quantities with Exponents

We cannot simplify by grouping two terms together unless they have the
same base and the same exponent. For example, we
cannot combine terms in expressions such as 5^{2} +12^{2} or 5^{3} +5^{4}. We can, however, simplify 4^{5} +4^{5} and 2*x*^{2} +5*x*^{2}. To
group two terms with the same base and the same exponent, add their
coefficients. Thus, 4^{5} +4^{5} = **1**(4)^{5} + **1**(4)^{5} = (1 + 1)(4)^{5} = 2(4)^{5} and 2*x*^{2} +5*x*^{2} = (2 + 5)*x*^{2} = 7*x*^{2}.

###
Multiplying Quantities with Exponents

We can multiply two quantities with exponents if they have the same
base. To multiply two quantities with the same base, multiply their
coefficients and add their exponents. For example, 4(5)^{5}×3(5)^{2} = (3×4)(5)^{5+2} = 12(5)^{7} and 5(2*x*)^{2}×6(2*x*)^{y} = (5×6)(2*x*)^{2+y} = 30(2*x*)^{2+y}.

###
Dividing Quantities with Exponents

We can divide two quantities with exponents if they have the same
base. To divide two quantities with the same base, divide their
coefficients and subtract their exponents. For example,
= (2)^{11-6} = 3(2)^{5} and
= *x*^{7-8} = *x*^{-1}.

###
Distributive Property of Exponents

If an exponent acts on single term in parentheses, we can
distribute the exponent over the term. For example, (2×5)^{2} = (2^{2})(5^{2}), (3*x*)^{6} = 3^{6}*x*^{6}, and 3(4*xy*)^{5} = 3(4^{5})*x*^{5}*y*^{5}.

*Be careful!* If an exponent acts on multiple terms in
parentheses (i.e. if there is a "+" or "-" sign in the parentheses),
it **cannot** be distributed: (5 + 3)^{2}≠5^{2} +3^{2} and (4*a* + *b*)^{2}≠*a*^{2} + *b*^{2}.

###
Taking a Power of a Power

Sometimes, the base will include an exponent, like in the expression
(2^{2})^{3}. If this is the case, multiply the exponent in the base by
the exponent which acts on the base: (2^{2})^{3} = 2^{2×3} = 2^{6}
and (*x*^{5})^{y} = *x*^{5×y} = *x*^{5y}.