Definition: we say that two radical expressions are like radicals if the index and radicand are the same.
Example: $4\sqrt{2}$ and $-2\sqrt{2}$ are like radicals since they both contain $\sqrt{2}.$
Example: $4\sqrt{3}$ and $-3\sqrt{2}$ are unlike radicals since they both contain different radicals.
Adding/Subtracting Like Radicals
Fact: Similar to like terms involving a variable, we can add/subtract like radicals.
Example: Subtract. $$4\sqrt{2}-2\sqrt{2}$$ Notice: The above example is similar to the subtraction problem $$4x-2x.$$
Examples
$2\sqrt{13}+5\sqrt{13}$
$2\sqrt{13}+5\sqrt{13}=7\sqrt{13}$
$2\sqrt[3]{3}+6\sqrt[3]{3}-9\sqrt[3]{3}$
$2\sqrt[3]{3}+6\sqrt[3]{3}-9\sqrt[3]{3}=-\sqrt[3]{3}$
$5\sqrt{7 q}+4\sqrt{7 q}$
$5\sqrt{7 q}+4\sqrt{7 q}=9\sqrt{7 q}$
$6\sqrt{19}+9\sqrt{13}+4\sqrt{19}+5\sqrt{13}$
$$
\begin{array}{lll}
\displaystyle 6\sqrt{19}+9\sqrt{13}+4\sqrt{19}+5\sqrt{13}&\displaystyle= 6\sqrt{19}+4\sqrt{19}+9\sqrt{13}+5\sqrt{13}&\mbox{}\\
\displaystyle &\displaystyle=10\sqrt{19}+14\sqrt{13} &\mbox{}\\
\end{array}
$$
Simplifying Radicals Before Adding
Fact: sometimes we need to simplify radicals before we can see that they are like radicals.
Example: Subtract $\sqrt[3]{704}-\sqrt[3]{88}$
$$
\begin{array}{lll}
\displaystyle \sqrt[3]{704}-\sqrt[3]{88}&\displaystyle=\sqrt[3]{704}-\sqrt[3]{88} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt[3]{\underline{2 \cdot 2 \cdot 2} \cdot \underline{2 \cdot 2 \cdot 2} \cdot 11}-\sqrt[3]{\underline{2 \cdot 2 \cdot 2} \cdot 11} &\mbox{}\\
\displaystyle &\displaystyle=2\cdot 2 \cdot \sqrt[3]{11}-2\sqrt[3]{ 11} &\mbox{}\\
\displaystyle &\displaystyle=4 \sqrt[3]{11}-2\sqrt[3]{ 11} &\mbox{}\\
\displaystyle &\displaystyle=2\sqrt[3]{ 11} &\mbox{}\\
\end{array}
$$
Example: Subtract $5\sqrt{32 n}-3\sqrt{8 n}$
$$
\begin{array}{lll}
\displaystyle 5\sqrt{32 n}-3\sqrt{8 n}&\displaystyle=5\sqrt{4^2\cdot 2 n}-3\sqrt{2^2 \cdot 2 n} &\mbox{}\\
\displaystyle &\displaystyle=5\sqrt{4^2}\sqrt{ 2 n}-3\sqrt{2^2}\sqrt{2 n} &\mbox{}\\
\displaystyle &\displaystyle=5\cdot 4 \cdot \sqrt{2 n}-3\cdot 2\cdot\sqrt{2 n} &\mbox{}\\
\displaystyle &\displaystyle=20\sqrt{ 2n}-6\sqrt{2 n} &\mbox{}\\
\displaystyle &\displaystyle=14\sqrt{ 2n} &\mbox{}\\
\end{array}
$$
More Examples
$2\sqrt{99 x^2}-4\sqrt{275 x^2}$
$$
\begin{array}{lll}
\displaystyle 2\sqrt{99 x^2}-4\sqrt{275 x^2}&\displaystyle= 2\sqrt{9x^2 \cdot 11}-4\sqrt{25x^2 \cdot 11}&\mbox{}\\
\displaystyle &\displaystyle=2\sqrt{9x^2}\sqrt{11}-4\sqrt{25x^2}\sqrt{11} &\mbox{}\\
\displaystyle &\displaystyle=2\cdot 3x\sqrt{11}-4\cdot 5x\sqrt{11} &\mbox{}\\
\displaystyle &\displaystyle=6x\sqrt{11}-20x\sqrt{11} &\mbox{}\\
\displaystyle &\displaystyle=-14x\sqrt{11} &\mbox{}\\
\end{array}
$$
$4\sqrt[5]{7 m}+9\sqrt[5]{7 m}+5\sqrt[5]{7 m}$
$4\sqrt[5]{7 m}+9\sqrt[5]{7 m}+5\sqrt[5]{7 m}=18\sqrt[5]{7 m}$
Fact of Life: Sometimes we also need to multiply radical expressions.
Recall Our Big Fact: A radical of a product is the product of the radicals. $$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$$ Example: Multiply and simplify the radical expression $\sqrt{13}(\sqrt{2}+\sqrt{7})$.
$$
\begin{array}{lll}
\displaystyle \sqrt{13}(\sqrt{2}+\sqrt{7})&\displaystyle= \sqrt{13}\sqrt{2}+\sqrt{13}\sqrt{7}&\mbox{}\\
\displaystyle &\displaystyle= \sqrt{13\cdot 2}+\sqrt{13\cdot 7}&\mbox{}\\
\displaystyle &\displaystyle= \sqrt{26}+\sqrt{91}&\mbox{}\\
\end{array}
$$
More Examples
$(9\sqrt{26 n})(2\sqrt{91 n})$
$$
\begin{array}{lll}
\displaystyle (9\sqrt{26 n})(2\sqrt{91 n})&\displaystyle=9\cdot 2\sqrt{26 n}\sqrt{91 n} &\mbox{}\\
\displaystyle &\displaystyle=18\sqrt{26 n \cdot 91 n} &\mbox{}\\
\displaystyle &\displaystyle=18\sqrt{2 \cdot 13 \cdot n \cdot 7 \cdot 13 \cdot n} &\mbox{}\\
\displaystyle &\displaystyle=18\sqrt{2 \cdot 7 \cdot 13 \cdot 13 \cdot n\cdot n } &\mbox{}\\
\displaystyle &\displaystyle=18\sqrt{14 \cdot (13n)^2 } &\mbox{}\\
\displaystyle &\displaystyle=18\sqrt{14} \sqrt{(13n)^2 } &\mbox{}\\
\displaystyle &\displaystyle=18\sqrt{14} \cdot 13n &\mbox{}\\
\displaystyle &\displaystyle=18\cdot 13n\sqrt{14} &\mbox{}\\
\displaystyle &\displaystyle=234n\sqrt{14} &\mbox{}\\
\end{array}
$$
$6\sqrt{t}(8\sqrt{ t}-4)$
$$
\begin{array}{lll}
\displaystyle 6\sqrt{t}(8\sqrt{ t}-4)&\displaystyle=6\sqrt{t}\cdot 8\sqrt{ t}-6\sqrt{t}\cdot 4 &\mbox{}\\
\displaystyle &\displaystyle=6\cdot 8\sqrt{t}\sqrt{ t}-6\cdot 4\sqrt{t} &\mbox{}\\
\displaystyle &\displaystyle=48t-24\sqrt{t} &\mbox{}\\
\end{array}
$$
$8\sqrt[3]{3}(6\sqrt[3]{18}+7\sqrt[3]{45})$
$$
\begin{array}{lll}
\displaystyle 8\sqrt[3]{3}(6\sqrt[3]{18}+7\sqrt[3]{45})&\displaystyle=8\sqrt[3]{3}\cdot 6\sqrt[3]{18}+8\sqrt[3]{3}\cdot 7\sqrt[3]{45} &\mbox{}\\
\displaystyle &\displaystyle=8\cdot 6\sqrt[3]{3}\sqrt[3]{18}+8\cdot 7\sqrt[3]{3}\sqrt[3]{45} &\mbox{}\\
\displaystyle &\displaystyle=48\sqrt[3]{3\cdot 9\cdot 2}+56\sqrt[3]{3\cdot 9\cdot 5} &\mbox{}\\
\displaystyle &\displaystyle=48\sqrt[3]{27\cdot 2}+56\sqrt[3]{27\cdot 5} &\mbox{}\\
\displaystyle &\displaystyle=48\sqrt[3]{27}\sqrt[3]{2}+56\sqrt[3]{27}\sqrt[3]{5} &\mbox{}\\
\displaystyle &\displaystyle=48\cdot 3\sqrt[3]{2}+56\cdot 3 \sqrt[3]{5} &\mbox{}\\
\displaystyle &\displaystyle=144\sqrt[3]{2}+168 \sqrt[3]{5} &\mbox{}\\
\end{array}
$$
Other Special Examples
$(\sqrt{2}+\sqrt{3})^2$ (hint: think FOIL)
$$
\begin{array}{lll}
\displaystyle (\sqrt{2}+\sqrt{3})^2&\displaystyle=(\sqrt{2}+\sqrt{3})(\sqrt{2}+\sqrt{3}) &\mbox{}\\
\displaystyle &\displaystyle=\underbrace{\sqrt{2}\sqrt{2}}_{\mbox{First}}+\underbrace{\sqrt{2}\sqrt{3}}_{\mbox{Outer}}+\underbrace{\sqrt{3}\sqrt{2}}_{\mbox{Inner}}+\underbrace{\sqrt{3}\sqrt{3}}_{\mbox{Last}} &\mbox{}\\
\displaystyle &\displaystyle=2+\sqrt{2\cdot 3}+\sqrt{3\cdot 2}+3 &\mbox{}\\
\displaystyle &\displaystyle=2+\sqrt{6}+\sqrt{6}+3 &\mbox{}\\
\displaystyle &\displaystyle=2+3+\sqrt{6}+\sqrt{6} &\mbox{}\\
\displaystyle &\displaystyle=5+2\sqrt{6} &\mbox{}\\
\end{array}
$$
$(\sqrt{2}+\sqrt{7})(\sqrt{2}-\sqrt{7})$ (hint: think FOIL)
$$
\begin{array}{lll}
\displaystyle (\sqrt{2}+\sqrt{7})(\sqrt{2}-\sqrt{7})&\displaystyle=\underbrace{\sqrt{2}\sqrt{2}}_{\mbox{First}}-\underbrace{\sqrt{2}\sqrt{7}}_{\mbox{Outer}}+\underbrace{\sqrt{7}\sqrt{2}}_{\mbox{Inner}}-\underbrace{\sqrt{7}\sqrt{7}}_{\mbox{Last}} &\mbox{}\\
\displaystyle &\displaystyle=2-\sqrt{2\cdot 7}+\sqrt{7\cdot 2}-7 &\mbox{}\\
\displaystyle &\displaystyle=2-\sqrt{14}+\sqrt{14}-7 &\mbox{}\\
\displaystyle &\displaystyle=2-7 &\mbox{}\\
\displaystyle &\displaystyle=-5 &\mbox{}\\
\end{array}
$$
$(\sqrt{3 z} + \sqrt{11 w})^2$ (hint: think FOIL)
$$
\begin{array}{lll}
\displaystyle (\sqrt{3z}+\sqrt{11w})^2&\displaystyle=(\sqrt{3z}+\sqrt{11w})(\sqrt{3z}+\sqrt{11w}) &\mbox{}\\
\displaystyle &\displaystyle=\underbrace{\sqrt{3z}\sqrt{3z}}_{\mbox{First}}+\underbrace{\sqrt{3z}\sqrt{11w}}_{\mbox{Outer}}+\underbrace{\sqrt{11w}\sqrt{3z}}_{\mbox{Inner}}+\underbrace{\sqrt{11w}\sqrt{11w}}_{\mbox{Last}} &\mbox{}\\
\displaystyle &\displaystyle=3z+\sqrt{3z\cdot 11w}+\sqrt{11w\cdot 3z}+11w &\mbox{}\\
\displaystyle &\displaystyle=3z+\sqrt{33wz}+\sqrt{33wz}+11w &\mbox{}\\
\displaystyle &\displaystyle=3z+11w+2\sqrt{33wz} &\mbox{}\\
\end{array}
$$
Please Remember...
Please Remember...
$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$ Cool. $\checkmark$
$\sqrt[n]{a+b}=\sqrt[n]{a}+\sqrt[n]{b}$ Not Cool!!! $\times$
$(\sqrt[n]{a}+\sqrt[n]{b})^{n}=a+b$ Not Cool!!! $\times$
$\sqrt[n]{a+b}=\sqrt[n]{a}+\sqrt[n]{b}$ Not Cool!!! $\times$
$(\sqrt[n]{a}+\sqrt[n]{b})^{n}=a+b$ Not Cool!!! $\times$