Simplifying Expressions Containing only Monomials
Monomials are algebraic expressions consisting of a single
term (though this term may contain subexpressions which do
contain more than one term). So here, we look at the
simplification of simple products, quotients, and powers –
in effect, we are reviewing and illustrating the laws of
exponents, but using algebraic expressions rather than simple
numerical expressions.
Symbolically, we have five distinct rules for combining
powers:
Usually simplification involves combining rules (iv) and (v)
with one or more of rules (i), (ii), or (iii). When division is
involved, rule (ii) brings in something like cancelling common
factors between the numerator and the denominator of a fraction.
We’ll show a few examples here, but our detailed
descriptions of methods for simplifying fractions must wait until
a later document in this series.
Remember that multiplication with simple numbers or symbols
representing simple numbers is commutative
– the order of the factors doesn’t matter:
a Â· b = b Â· a
Example 1:
Simplify (3x^{ 2 })(5x^{ 3} )
solution:
We can rewrite this expression in detail as
(3x^{ 2 })(5x^{ 3} ) = (3)(x^{ 2 })(5)(x^{
3} )
the product of four distinct factors. Now
(3)(x^{ 2 })(5)(x^{ 3} ) = (3)(5)(x^{ 2 })(x^{
3} )
by rearranging the order of the factors, which leaves the
result unchanged since the multiplication here is commutative.
Now
(3)(x^{ 2 })(5)(x^{ 3} ) = (3 Â· 5)(x^{ 2 }Â·
x^{ 3} ) = 15x^{ 2 + 3} = 15x^{ 5}
So, we conclude that
(3x^{ 2 })(5x^{ 3} ) = 15x^{ 5}
Example 2:
Simplify (2x^{ 3 })^{2 }(3x^{ 3} )^{4
}
solution:
Using property (iv)
(2x^{ 3 })^{2 }= (2)^{2} (x^{ 3 })^{2}
= 4x^{ 3 Â· 2 }= 4x^{ 6}
and
(3x^{ 3 })^{4 }= (3)^{4} (x^{ 3 })^{4}
= 81x^{ 3 Â· 4 }= 81x^{ 12}
So,
(2x^{ 3 })^{2 }(3x^{ 3} )^{4 }=
(4x^{ 6} )(81x^{ 12}) = (4 Â· 81)(x^{
6 }Â· x^{ 12} ) = 324x^{ 6 + 12} = 324x^{
18}
Example 3:
Simplify (3x^{ 2 })^{5 }(2x^{ 4} )^{2
}
solution:
Care must be taken with the minus sign here. Again, applying
property (iv), we get
(3x^{ 2 })^{5 }= (3)^{5 }(x^{ 2 })^{5
}= 243x ^{10}
and
(2x^{ 4} )^{2 }= (2)^{2 }(x^{ 4}
)^{2 }= 4x^{ 8}
So
(3x^{ 2 })^{5 }(2x^{ 4} )^{2 }=
(243x ^{10} )(4x^{ 8} ) = 972x ^{10+8}
= 972x ^{18}
