Rationalizing an algebraic expression means rewriting it so that a radical is removed from a denominator
or, less commonly, from a numerator. The value of the expression does not change; only its form changes.
1. Why rationalize?
A fraction such as
\[
\begin{aligned}
\frac{1}{\sqrt{2}}
\end{aligned}
\]
has a radical in the denominator. In many algebra courses, the preferred equivalent form is:
\[
\begin{aligned}
\frac{\sqrt{2}}{2}.
\end{aligned}
\]
The denominator is now rational.
2. Single radical denominator
For a denominator containing one square root, multiply by the same square root:
\[
\begin{aligned}
\frac{1}{\sqrt{a}}
\cdot
\frac{\sqrt{a}}{\sqrt{a}}
&=
\frac{\sqrt{a}}{a}.
\end{aligned}
\]
This works because:
\[
\begin{aligned}
\sqrt{a}\cdot\sqrt{a}
=
a.
\end{aligned}
\]
3. Conjugates
When a denominator has two terms, such as \(a+\sqrt{b}\), use the conjugate \(a-\sqrt{b}\).
\[
\begin{aligned}
(a+\sqrt{b})(a-\sqrt{b})
&=
a^2-b.
\end{aligned}
\]
The radical disappears because the middle terms cancel.
4. Worked example: \(4/(\sqrt{5}-\sqrt{3})\)
Start with:
\[
\begin{aligned}
\frac{4}{\sqrt{5}-\sqrt{3}}.
\end{aligned}
\]
The conjugate of the denominator is:
\[
\begin{aligned}
\sqrt{5}+\sqrt{3}.
\end{aligned}
\]
Multiply top and bottom by this conjugate:
\[
\begin{aligned}
\frac{4}{\sqrt{5}-\sqrt{3}}
\cdot
\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}
&=
\frac{4(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})}.
\end{aligned}
\]
Simplify the denominator using difference of squares:
\[
\begin{aligned}
(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})
&=
(\sqrt{5})^2-(\sqrt{3})^2\\
&=
5-3\\
&=
2.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\frac{4(\sqrt{5}+\sqrt{3})}{2}
&=
2(\sqrt{5}+\sqrt{3}).
\end{aligned}
\]
Final answer:
\[
\begin{aligned}
\boxed{
\frac{4}{\sqrt{5}-\sqrt{3}}
=
2(\sqrt{5}+\sqrt{3})
}.
\end{aligned}
\]
5. Denominators with a rational term and a radical term
For a denominator like \(3+2\sqrt{2}\), use the conjugate \(3-2\sqrt{2}\):
\[
\begin{aligned}
(3+2\sqrt{2})(3-2\sqrt{2})
&=
3^2-(2\sqrt{2})^2\\
&=
9-8\\
&=
1.
\end{aligned}
\]
So:
\[
\begin{aligned}
\frac{1}{3+2\sqrt{2}}
&=
3-2\sqrt{2}.
\end{aligned}
\]
6. Rationalizing a numerator
Sometimes a problem asks you to remove a radical from the numerator instead.
The same conjugate idea applies.
\[
\begin{aligned}
\frac{\sqrt{5}-\sqrt{3}}{4}
\cdot
\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}
&=
\frac{2}{4(\sqrt{5}+\sqrt{3})}.
\end{aligned}
\]
The numerator is rationalized, although the denominator now contains radicals.
7. Simplifying radicals
A radical should be simplified when the radicand contains a perfect-square factor.
\[
\begin{aligned}
\sqrt{12}
&=
\sqrt{4\cdot 3}\\
&=
2\sqrt{3}.
\end{aligned}
\]
This calculator simplifies numeric square roots before displaying the final answer.
8. De-nesting selected nested radicals
Some nested radicals can be rewritten as a sum or difference of simpler square roots.
For example:
\[
\begin{aligned}
\sqrt{3+2\sqrt{2}}
&=
\sqrt{2}+1.
\end{aligned}
\]
The reason is:
\[
\begin{aligned}
(\sqrt{2}+1)^2
&=
2+2\sqrt{2}+1\\
&=
3+2\sqrt{2}.
\end{aligned}
\]
In general, the calculator looks for:
\[
\begin{aligned}
\sqrt{a\pm b\sqrt{c}}
=
\sqrt{m}\pm\sqrt{n},
\end{aligned}
\]
where \(m+n=a\) and \(mn=b^2c/4\).
9. Why the conjugate works
The main identity is the difference of squares:
\[
\begin{aligned}
(u-v)(u+v)
=
u^2-v^2.
\end{aligned}
\]
If \(u\) and \(v\) contain square roots, their squares often become rational numbers.
That is why multiplying by the conjugate removes the radical from the selected part.
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Multiplying only the denominator by the conjugate and forgetting the numerator.
- Using the same sign instead of the opposite sign in the conjugate.
- Forgetting that \((\sqrt{a})^2=a\).
- Not simplifying the final radical expression.
- Trying to use the simple cover/conjugate method on expressions with too many terms.
- Assuming every nested radical can be de-nested nicely.
Key idea: multiply by a form of 1 using the conjugate, so the radical terms cancel from the denominator or numerator.
