RBQE - Operations with Radical Expressions (Lesson)

Operations with Radical Expressions

Examples

$LaTeX: 3\sqrt[2]{2}\cdot4\sqrt[2]{18}\\=12\sqrt[2]{36}\\=12\cdot6\\=72$ $3\sqrt[2]{2}\cdot4\sqrt[2]{18}\\=12\sqrt[2]{36}\\=12\cdot6\\=72$
$LaTeX: \sqrt[2]{\frac{2}{3}}\cdot\sqrt[2]{\frac{5}{12}}\\=\sqrt[2]{\frac{10}{36}}\\=\frac{\sqrt[2]{10}}{6}$ $\sqrt[2]{\frac{2}{3}}\cdot\sqrt[2]{\frac{5}{12}}\\=\sqrt[2]{\frac{10}{36}}\\=\frac{\sqrt[2]{10}}{6}$

in order to ensure a radical does not remain in the denominator, we use a process called rationalizing. In this process, we multiply by an appropriate value of 1.

Watch this video on rationalizing to see how it works!

Multiplying, Dividing, and Rationalizing Radicals Practice

$LaTeX: \sqrt[2]{\frac{4}{9}}\cdot\sqrt[2]{\frac{18}{4}}$ $\sqrt[2]{\frac{4}{9}}\cdot\sqrt[2]{\frac{18}{4}}$
$LaTeX: \sqrt[2]{\frac{8}{11}}\cdot\sqrt[2]{\frac{22}{32}}$ $\sqrt[2]{\frac{8}{11}}\cdot\sqrt[2]{\frac{22}{32}}$
$LaTeX: \frac{\sqrt[2]{20}}{\sqrt[2]{15}}$ $\frac{\sqrt[2]{20}}{\sqrt[2]{15}}$
$LaTeX: \sqrt[2]{\frac{7}{3}}\cdot\sqrt[2]{\frac{3}{28}}$ $\sqrt[2]{\frac{7}{3}}\cdot\sqrt[2]{\frac{3}{28}}$
$LaTeX: \frac{\sqrt[2]{2}}{\sqrt[2]{5}}$ $\frac{\sqrt[2]{2}}{\sqrt[2]{5}}$

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

When adding and subtracting radicals, we must be sure they are like radicals, meaning that the radicands (value under the radical) must be the same. See below:

$LaTeX: 2\sqrt[2]{3}=\sqrt[2]{12}$ $2\sqrt[2]{3}=\sqrt[2]{12}$
$LaTeX: \sqrt[2]{63}=3\sqrt[2]{7}$ $\sqrt[2]{63}=3\sqrt[2]{7}$
$LaTeX: 2\sqrt[2]{5}=\sqrt[2]{20}$ $2\sqrt[2]{5}=\sqrt[2]{20}$
$LaTeX: 2\sqrt[2]{2}=\sqrt[2]{8}$ $2\sqrt[2]{2}=\sqrt[2]{8}$

Examples

$LaTeX: \sqrt[2]{24}+3\sqrt[2]{6}\:=\:\sqrt[2]{4\cdot6}+3\sqrt[2]{6}=2\sqrt[2]{6}+3\sqrt[2]{6}=5\sqrt[2]{6}$ $\sqrt[2]{24}+3\sqrt[2]{6}\:=\:\sqrt[2]{4\cdot6}+3\sqrt[2]{6}=2\sqrt[2]{6}+3\sqrt[2]{6}=5\sqrt[2]{6}$

Another example of adding and subtracting with radicals

Notice that in the second example, we could not add negative fifteen square root of six plus three square root of two because they are not like radicals.

Add and Subtract Radicals Practice

Try these problems!

$LaTeX: 5\sqrt[2]{28}+2\sqrt[2]{7}-\sqrt[2]{14}$ $5\sqrt[2]{28}+2\sqrt[2]{7}-\sqrt[2]{14}$
$LaTeX: 3\sqrt[2]{32}-4\sqrt[2]{2}$ $3\sqrt[2]{32}-4\sqrt[2]{2}$
$LaTeX: 5\sqrt[2]{80}-12\sqrt[2]{5}$ $5\sqrt[2]{80}-12\sqrt[2]{5}$
$LaTeX: -3\sqrt[2]{72}+6\sqrt[2]{52}-7\sqrt[2]{128}$ $-3\sqrt[2]{72}+6\sqrt[2]{52}-7\sqrt[2]{128}$

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

IMAGES CREATED BY GAVS