Radical Expressions
Lesson · Intermediate
Algebra
Let x and y be nonnegative integers.
If
where d is a positive squarefree integer and k is a positive integer, then
for some nonnegative integers a and b.
In particular,
Equivalently,
Example
If
then
where a and b are nonnegative integers and
For nonnegative real numbers a and b,
and for a ≥ 0 and b > 0,
For a ≥ 0 and b ≥ 0,
For example,
For example,
More generally,
and
provided the denominators are nonzero and all radicals are real.
Suppose
Squaring,
Hence
so
is rational. Since xy is an integer, this implies that its square root is an integer, so xy is a perfect square.
If x = 0, then y = k²d, so we may take a = 0 and b = k. Similarly, if y = 0, we may take a = k and b = 0. Thus the conclusion holds in these cases.
Now suppose x and y are both positive. Write
where a and b are positive integers and r and s are positive squarefree integers. Since xy is a perfect square,
is a perfect square. Because r and s are squarefree,
Thus
Hence
Since r and d are positive squarefree integers,
Therefore
and
Let x and y be nonnegative integers satisfying
Find all possible ordered pairs (x, y).
Since
we must have
where
Thus
For
we obtain
Let x and y be positive integers satisfying
and
Find x and y.
Write
where
Also,
so
Since a + b = 12,
Therefore
Hence
Simplify
Multiply the numerator and denominator by the conjugate:
Therefore
Find all positive integers n such that
is an integer.
Suppose
Since the right-hand side is k times the square root of 1, the main property gives
for positive integers a and b.
Thus
so
The positive factor pairs give
Hence
respectively, and therefore
If
for nonnegative integers x, y, and k, then take d = 1.
Therefore
and hence
The idea extends to several radicals.
If
where x₁, ..., xₙ are nonnegative integers, d is a positive squarefree integer, and k is a nonnegative integer, then every nonzero radical must have the same squarefree part d.
Thus
for nonnegative integers aᵢ, and
Before comparing or combining radicals, write each radical in the form
where d is squarefree.
For example,
The original radicands 12 and 75 are different, but their squarefree parts are the same.
This remark is important for the main theorem.
In general,
For example,
but
For real a,
not necessarily a.
For example,
You cannot simplify
to
Only radicals with the same squarefree part can be combined into a single like radical term.
Over the real numbers, this property requires