How can we simplify expressionsusing exponents?
The following are ways to solve simple mathematical expressions with the same base:
To multiply two numbers with different exponents, add the exponents:
(x^2 )(x^4 )
(xx)(xxxx)
xxxxxx
x^2 ^4
x^6
To raise a number with a power, multiply the exponents:
(x^3 )^3
(x^3 )(x^3 )(x^3 )
(xxx)(xxx)(xxx)
xxxxxxxxx
x(33)
x^9What common mistakeis often made in exponential equations?
Unlike multiplication, exponents do not “distribute” over addition. For example, (2
5)^2 does notmean 2^2 52 4 25 29. In this case, you add the numbers in the
parentheses first, then square that number. The trueanswer to this equation is (2
5)^2 (7)^2 49.
What is a basein mathematics?
The term “base” has many meanings in the English language, including several that
apply to the field of mathematics. When talking about sets, bases are the open sets
whose union forms an abstract entity called a topological space. In geometry, the base
represents the side of a polygon or polyhedron that is perceived as its bottom; when
referring to an isosceles triangle, the base is the side that differs in length from the
other two (thus, the base angles include the side that is thought of as the base). Alge-
braists also use the term base to describe either the number used with an exponent to
create a power, such as 3^4 81; or to write the same number as a subscript to a loga-
rithm, such as log 381 4. (For more information about logarithms, see “Algebra.”)
One of the more familiar uses of the term base in mathematics deals with our
numbering system, in which a base is a natural number whose powers are added to
produce a specific number. For example, using 10 as a base, the number 2583.789 is
actually (2 103 ) (5 102 ) (8 101 ) (3 100 ) (7 10 ^1 ) (8 10 ^2 )
(9 10 ^3 ). 87