What does
"Rationalizing
the
Denominator"
mean?
Rationalizing
the
Denominator
simply means to
remove
all radicals
from the
denominator
of a fraction
without
changing the
value of the
fraction.
When the
denominator is
rationalized,
the original
fraction is
converted to
the simplest
equivalent
fraction which
does not have
radicals in the
denominator.
By removing all
radicals from
the
denominator,
all numbers in
the denominator
will be
converted
rational
numbers (hence
the term,
"Rationalizing
the
Denominator").
For
example,
the
following
fraction
has a
square
root in
the
denominator:
(Note
that the square
root of 2 is an
irrational
number - a
non-terminating
decimal without
a repeating
pattern.)
Rationalizing
the
Denominator
means the
fraction
will be
rewritten
as the
simplest
equivalent
fraction
which does
not have
radicals
in the
denominator:
(Note
that the
denominator is
now the
rational number
2.)
Why
Rationalize the
Denominator of
a
fraction?
Why not leave
the denominator
alone?
What difference
does it make?
What is the
point of
Rationalizing
the
Denominator?
Rationalizing
the
Denominator
is the
standard
way of
simplifying
fractions
so that
they
(fractions)
can be
readily
understood
and easily
compared
with other
fractions.
The
keys
to
understanding
why
Rationalizing
the
Denominator
is
important
are:
simplification
and
standardization:
Example of
how the
use of
integers
is
standardized
Suppose
you were
telling a
friend you
had 5
apples
yesterday,
but today
you only
have 4
apples
because
you ate 1
apple.
How would
you
present
the
numbers 5
and 4?
Statement
1:
"I
had 5
apples
yesterday.
However, I
ate 1
apple and
now I have
only 4
apples."
This
statement
is
straightforward
and makes
it easy to
compare
how many
apples you
had
yesterday
with the
number you
have today
(the
numbers 5
and 4).
But that
is not the
only way
you could
communicate
the
numbers 5
and 4.
Statement
2:
"I
had 6
minus 1
apples
yesterday;
but, only
100 minus
96 apples
today."
This
statement
makes it
difficult
to compare
how many
apples you
had
yesterday
with the
number you
have today
(the
numbers 5
and 4)
without
completing
a
subtraction
problem.
Statements
1 & 2 are
mathematically
equivalent.
Each shows
exactly
how many
apples you
have on
both days.
However,
Statement
1 presents
the
numbers 5
and 4 as
single
integers .
. . the
standard
for
clarity
and ease
of
comparison.
Example of
how the
use of
fractions
is
standardized
Reducing a
fraction
to its
simplest
form makes
it easier
to
understand
and
compare
with other
fractions
in the
majority
of cases.
For
example,
the
following
four
fractions
have
exactly
the same
numerical
value:
However,
if you
were
telling a
friend you
have half
an apple,
you would
probably
use the
fraction
because
it is
easy
to
understand
(it
is
the
most
reduced
form
of
the
fraction).
But you
could use
any of the
four
fractions
listed.
You could
say you
had
of
an apple.
Fractions
are
generally
reduced to
the
simplest
form
because it
makes them
easier to
understand.
Rationalizing
the
denominator
of a
fraction
simplifies
and
standardizes
fractions
in the
same
way
Removing
all
radicals
from the
denominator
of every
fraction
is a
convention
which
allows
fractions
to be
compared
more
easily.
The value
of the
fraction
is not
changed,
but it is
easier to
understand
and
compare
with other
fractions.
Is it
ever necessary
to rationalize
the numerator
of a fraction
instead of a
denominator?
Yes. When
studying
calculus,
it will be
necessary
to find
limits. In
this case,
rationalizing
the
numerator
of a
fraction
rather
than the
denominator
is part of
the
process.
However,
the basic
technique
for
rationalization
either the
denominator
or
numerator
is the
same.
How to
rationalize the
denominator of
a
fraction
Video
demonstrating
How to
Rationalize the
Denominator
-
click
here
The
denominator
is
a
monomial
(1
term).
To
rationalize
the
denominator,
(1)
multiply
the
denominator
by a
number (or
expression)
which will
remove the
radical
from the
denominator.
(2)
Multiply
the
numerator
by the
same
number (or
expression).
Rationalize
the
denominator
of:
-
simplifying
radicals
-
The
final answer
is:
Rationalize
the
denominator
of:
-
simplifying
radicals
-
The
final answer
is:
Rationalize
the
denominator
of:
-
simplifying
radicals
-
The
final answer
is:
Rationalize
the
denominator
of:
-
simplifying
radicals
-
The
final answer
is:
The
denominator
is
a
binomial
(2
terms).
To
rationalize
the
denominator,
(1)
multiply
the
denominator
by an
expression
which is
the
conjugate
of the
denominator.
(2)
Multiply
the
numerator
by the
same
expression.
The
conjugate
of
"A+B"
is
"A-B".
The
conjugate
of
"A-B"
is
"A+B".
Rationalize
the
denominator
of:
-
simplifying
radicals
-
The
final answer
is:
Rationalize
the
denominator
of:
-
simplifying
radicals
-
The
final answer
is:
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