\(a\)	\(x^{2}=a\)
\(a>0\)	\(x=\pm\sqrt{a}\)
\(a=0\)	\(x=0\)
\(a<0\)	None

Calculation

Let \(a>0\) and \(b>0\).

\((\sqrt{a})^{2}=a\)

\(\sqrt{a^{2}}=|a|\)

\(m\sqrt{a}+n\sqrt{a}=(m+n)\sqrt{a}\)

\(\sqrt{a}\sqrt{b}=\sqrt{ab}\)

\(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\)

Remark: \(\sqrt{a}\pm\sqrt{b}\not=\sqrt{a\pm b}\)

Rationalization of the Denominator

Let \(a>0\), \(b>0\) and \(a\not= b\).

Calculation	Formula
\(\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{\sqrt{a}\sqrt{a}}=\frac{\sqrt{a}}{a}\)	\(a\cdot a=a^{2}\)
\(\frac{1}{\sqrt{a}\pm\sqrt{b}}=\frac{\sqrt{a}\mp\sqrt{b}}{(\sqrt{a}\pm\sqrt{b})(\sqrt{a}\mp\sqrt{b})}=\frac{\sqrt{a}\mp \sqrt{b}}{a-b}\)	\((a+b)\cdot (a-b) =a^{2}-b^{2}\)
\(\frac{1}{\sqrt[3]{a}\pm\sqrt[3]{b}}=\frac{\sqrt[3]{a^{2}}\mp\sqrt[3]{a}\sqrt[3]{b}+\sqrt[3]{b}}{(\sqrt[3]{a}\pm\sqrt[3]{b})(\sqrt[3]{a^{2}}\mp\sqrt[3]{a}\sqrt[3]{b}+\sqrt[3]{b})}=\frac{\sqrt[3]{a^{2}}\mp\sqrt[3]{a}\sqrt[3]{b}+\sqrt[3]{b}}{a\pm b}\)	\((a+b)(a^{2}-ab+b^{2})=a^{3}+b^{3}\) \((a-b)(a^{2}+ab+b^{2})=a^{3}-b^{3}\)

Example: Rationalize \(\frac{2}{\sqrt[3]{5}-\sqrt[3]{3}}\).

Let \(a=\sqrt[3]{5}\) and \(b=\sqrt[3]{3}\). Then

$$\frac{2}{\sqrt[3]{5}-\sqrt[3]{3}}=\frac{2(a^{2}+ab+b^{2})}{(a-b)(a^{2}+ab+b^{2})}=\frac{2(a^{2}+ab+b^{2})}{a^{3}-b^{3}}=\frac{2(\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9})}{2}=\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}$$

Double Root Sign

Let \(a>0\), \(b>0\) and \(a>b\).