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 .

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}
.

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}
.

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}
.

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}
.

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}
.

We cannot multiply quantities if they do not have the same base. However, we can often use the "power of a power" rule to write two quantities as quantities with a common base. If the two bases are powers of a common base (for example, 4 and 32 are both powers of 2 ), rewrite each base as a power of the common base. Then uses the power of a power rule to combine the exponents of each base. Finally, use the multiplication rule and add the exponents.

Example:
9^{5}×27^{4} =
?

9^{5} = (3^{2})^{5} = 3^{1}0
;
27^{4} = (3^{3})^{4} = 3^{1}2

9^{5}×27^{4} = 3^{1}0×3^{1}2 = 3^{10+12} = 3^{2}2
.

**Addition**:
*cx*
^{a} + *dx*
^{a} = (*c* + *d* )*x*
^{a}

**Subtraction**:
*cx*
^{a} - *dx*
^{a} = (*c* - *d* )*x*
^{a}

**Multiplication**:
*cx*
^{a}×*dx*
^{b} = (*cd* )*x*
^{a+b}

**Division**:
= ()*x*
^{a-b}

**Distribution**:
(*cd* )^{a} = *c*
^{a}
*d*
^{a}

**Power of a Power**:
*c*(*x*
^{a})^{b} = *cx*
^{ab}