To simplify radicals means to reduce the expression under the radical sign. Like reducing fractions: \(\frac{24}{18}=\frac{4}{3}\), radicals have a similar property used to simplify called the Product Rule.
In words, the product of two \(n^{th}\) roots will produce another \(n^{th}\) root. In a formula, it looks like: \(\sqrt[n]{A}\cdot\sqrt[n]{B}=\sqrt[n]{A\cdot B}\).
NOTE: To multiply radicals, the index on each radical must be the same.
Multiply the following radicals using the Product Rule.
| (a) \(\sqrt[3]{2}\cdot\sqrt[3]{7}\) | (b) \(\sqrt[6]{8r^2}\cdot\sqrt[6]{2r^3}\) |
| \(\implies\sqrt[3]{2\cdot 7}\) | \(\implies\sqrt[6]{(8r^2)(2r^3)}\) |
| \(\implies\sqrt[3]{14}\) | \(\implies\sqrt[6]{16r^5}\) |
| (c) \(\sqrt[5]{9y^2x}\cdot\sqrt[5]{8xy^2}\) | (d) \(\sqrt{7}\cdot\sqrt[3]{5}\) |
| \(\implies\sqrt[5]{(9y^2x)(8xy^2)}\) | Cannot be multiplied since the indices are different |
| \(\implies\sqrt[5]{72x^2y^4}\) |
Typically, we will use the Product Rule backwards: \(\sqrt[3]{64\cdot3}\implies\sqrt[3]{64}\cdot\sqrt[3]{3}\) for simplification.
There is a Quotient Rule for radicals as well, however, it is not used as heavily as the Product Rule. The following example illustrates the use:
$$\sqrt[3]{-\frac{8}{125}}\implies\frac{\sqrt[3]{-8}}{\sqrt[3]{125}}\implies\frac{-2}{5}$$
A radical is simplified provided the following conditions are meet:
| Conditions for a Simplified Radical |
|---|
| 1. Exponents under the radical are NOT greater than or equal to the index. |
| 2. There are no fractions under the radical. |
| 3. There are no radicals in the denominator of a fraction. |
| 4. Exponents under the radical and the index have a GCF = 1. |
To begin the simplifcation, you must know your perfect squares: \({1,4,9,16,25,36,49, . . .}\), cubes: \({1,8,27,64,125, . . .}\) and a few \(4^{th}\) roots: \({1,16,81,256,625, . . .}\). Any \(n^{th}\) root greater than 4 can be calculated by a calculator. Most problems will be simply finding the largest root that matches the index:
Simplify the following roots:
| (a) \(\sqrt[3]{54}\) | (b) \(\sqrt{108}\) | (c) \(-\sqrt[4]{243}\) | |
|---|---|---|---|
| \(\begin{array}{l} \implies \sqrt[3]{27\cdot2} \\ \implies\sqrt[3]{27}\cdot\sqrt[3]{2} \\ \implies3\cdot\sqrt[3]{2} \\ \implies3\sqrt[3]{2} \end{array}\) | Method 1 \begin{array}{l} \implies\sqrt{9\cdot12} \\ \implies\sqrt{9}\cdot\sqrt{12}\Rightarrow\sqrt{12}=\sqrt{\color{red}{4\cdot3}} \\ \implies3\cdot\sqrt{\color{red}{4}}\cdot\sqrt{\color{red}{3}} \\ \implies3\cdot2\cdot\sqrt{3} \\ \implies6\sqrt{3} \end{array} | Method 2 \begin{array}{l} \implies\sqrt{36\cdot3} \\ \implies\sqrt{36}\cdot\sqrt{3} \\ \implies6\cdot\sqrt{3} \\ \implies 6\sqrt{3} \end{array} | \(\begin{array}{l} \implies-\sqrt[4]{81\cdot3} \\ \implies-\sqrt[4]{81}\cdot\sqrt[4]{3} \\ \implies-3\cdot\sqrt[4]{3} \\ \implies -3\sqrt[4]{3} \end{array}\) |
The following examples illustrate each simplification condition above.
Simplify \(\sqrt{40ab^3}\).
Since \(40=8\cdot5\) and \(8=(2)^3\), under the radical are exponents of a \(3\) which are greater than the index of \(2\).
| \(\begin{array}{l} \sqrt{(2)^{\color{red}{3}}(5)ab^{\color{red}{3}}} & \implies \sqrt{(2)^3}\cdot\sqrt{5}\cdot\sqrt{a}\cdot\sqrt{b^3} \\ & \implies \sqrt{(2)^2}\cdot\sqrt{2}\cdot\sqrt{5}\cdot\sqrt{a}\cdot\sqrt{b^2}\cdot\sqrt{b} \\ & \implies 2\cdot\sqrt{2}\cdot\sqrt{5}\cdot\sqrt{a}\cdot|b|\cdot\sqrt{b} \\ & \implies 2|b|\cdot\sqrt{2\cdot5ab} \\ & \implies 2|b|\sqrt{10ab} \end{array}\) |
Simplify \(\sqrt{\frac{6}{5}}\).
Sometimes, one application of the Quotient Rule will simplify the radical, but here is does not.
$$\sqrt{\frac{6}{5}}\implies\frac{\sqrt{6}}{\sqrt{5}}$$
Notice, there is still a radical in the denominator, so this is NOT simplified completely. The topic
Multiply & Divide Radicals will show the next steps to simplify. For now, this is as simplified as far as it can be.
Simplify: \(\sqrt[6]{2x^3}\)
First use the Product Rule: \(\sqrt[6]{2x^3}\implies\sqrt[6]{2}\cdot\sqrt[6]{x^3}\). The first radical is simplified. The second radical has an index of \(6\) and an exponent of \(3\), the GCF = \(3\), not \(1\), so this is NOT simplified. Use the rational exponents form for radicals:
| \(\begin{array}{l} \sqrt[6]{2}\cdot\sqrt[6]{x^3} & \implies \sqrt[6]{2}\cdot\left(x^3\right)^{1/6} \\ & \implies \sqrt[6]{2}\cdot\left(x\right)^{3/6}\Rightarrow\color{red}{\frac{3}{6}=\frac{1}{2}} \\ & \implies \sqrt[6]{2}\cdot\left(x\right)^{1/2} \\ & \implies \sqrt[6]{2}\cdot\sqrt{x}\end{array}\) |
This is now simplified, since the indices are not the same, the Product Rule does not apply.