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Roots and Radicals: (lesson 1 of 3)

Simplifying Radical Expressions

To begin the process of simplifying radical expression, we must introduce the product and quotient rule for radicals

Product and quotient rule for radicals

Product Rule for Radicals: If $ \sqrt[n]{a} $ and $ \sqrt[n]{b} $ are real numbers and $n$ is a natural number, then $$ \large{\color{blue}{\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}}} $$

That is, the product of two radicals is the radical of the product.

Example 1 - using product rule

$$ a) \sqrt{\color{red}{6}} \cdot \sqrt{\color{blue}{5}} = \sqrt{\color{red}{6} \cdot \color{blue}{5}} = \sqrt{30} $$ $$ b) \sqrt{\color{red}{5}} \cdot \sqrt{\color{blue}{2ab}} = \sqrt{\color{red}{5} \cdot \color{blue}{2ab}} = \sqrt{10ab} $$
$$ c) \sqrt[4]{\color{red}{4a}} \cdot \sqrt[4]{\color{blue}{7a^2b}} = \sqrt[4]{\color{red}{4} \cdot \color{blue}{7a^2b}} = \sqrt[4]{28a^2b} $$

Quotient Rule for Radicals: If $ \sqrt[n]{a} $ and $ \sqrt[n]{b} $ are real numbers, $ b \ne 0 $ and $ n $ is a natural number, then $$ \color{blue}{\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[\large{n}]{\frac{a}{b}}} $$

That is, the radical of a quotient is the quotient of the radicals.

Example 2 - using quotient rule

$$ a) \sqrt{\frac{\color{red}{5}}{\color{blue}{36}}} = \frac{ \sqrt{\color{red}{5}} } { \sqrt{\color{blue}{36}} } = \frac{\sqrt{6}}{5} $$ $$ b) \sqrt[3]{\frac{\color{red}{a}}{\color{blue}{27}}} = \frac{ \sqrt[3]{\color{red}{a}} }{ \sqrt[3]{\color{blue}{27}} } = \frac{\sqrt[3]{a}}{3} $$
$$ c) \sqrt[4]{\frac{\color{red}{81}}{\color{blue}{64}}} = \frac{\sqrt[4]{\color{red}{81}} }{\sqrt[4]{\color{blue}{64}} } = \frac{3}{2} $$

Exercise 1: Simplify radical expression

Level 1

$$ \color{blue}{\sqrt5 \cdot \sqrt{15} \cdot{\sqrt{27}}} $$ $ 5\sqrt{27} $
$ 30 $
$ 45 $
$ 30\sqrt2 $

Level 2

$$ \color{blue}{\sqrt{\frac{32}{64}}} $$ $ \frac{\sqrt2}{2} $
$ 2\sqrt2 $
$ \frac{2}{\sqrt2} $
$ 2 $

Simplifying Roots of Numbers

Example 3: Simplify $ \sqrt{18} $

Solution:

Step 1: We need to find the largest perfect square that divides into 18. Such number is 9.

Step 2:Write 18 as the product of 2 and 9. ( 18 = 9 * 2 )

Step 3:Use the product rule: $ \sqrt{18} = \sqrt{\color{red}{9} \cdot \color{blue}{2}} = \sqrt{\color{red}{9}} \cdot \sqrt{\color{blue}{2}} = 3\sqrt{2} $

Example 4: Simplify $ \sqrt{108} $

Solution:

Step 1:Again,we need to find the largest perfect square that divides into 108. Such number is 36.

Step 2:Write 108 as the product of 36 and 3. ( 108 = 36 * 3 )

Step 3:Use the product rule: $ \sqrt{108} = \sqrt{\color{red}{36} \cdot \color{blue}{3}} = \sqrt{\color{red}{36}} \cdot \sqrt{\color{blue}{3}} = 6\sqrt{3} $

Example 5: Simplify $\sqrt{15}$

Solution:

No perfect square divides into 15, so $\sqrt{15} $ cannot be simplified

Example 6: Simplify $ \sqrt[3]{24} $

Solution:

Step 1: Now, we need to find the largest perfect cube that divides into 24. Such number is 8.

Step 2:Write 24 as the product of 8 and 3. ( 24 = 8 * 3 )

Step 3:Use the product rule: $ \sqrt[3]{24} = \sqrt[3]{\color{red}{8} \cdot \color{blue}{3}} = \sqrt[3]{\color{red}{8}} \cdot \sqrt[3]{\color{blue}{3}} = 2\sqrt[3]{3} $

Exercise 2: Simplify expression

Level 1

$$ \color{blue}{\sqrt{128}} $$ $ 4\sqrt2 $
$ \sqrt2 $
$ 8\sqrt2 $
$ \sqrt6 $

Level 2

$$ \color{blue}{\sqrt[\large{3}]{128}} $$ $ 2\sqrt[\large{3}]{2} $
$ 6\sqrt[\large{3}]{2} $
$ 4\sqrt[\large{3}]{2} $
$ 12\sqrt[\large{3}]{2} $

Simplifying Radicals Involving Variables

Examples 7: In this examples we assume that all variables represent positive real numbers.

$$ \begin{aligned} a) & \sqrt{4 \cdot a^3} = \sqrt{\color{red}{4} \cdot \color{blue}{a^2} \cdot a} = \sqrt{\color{red}{4}} \cdot \sqrt{\color{blue}{a^2}} \cdot \sqrt{a} = 2a\sqrt{a} \\ b) & \sqrt{9 \cdot 7^3} = \sqrt{\color{red}{9} \cdot \color{blue}{(b^3)^2} \cdot b} = \sqrt{\color{red}{9}} \cdot \sqrt{\color{blue}{(b^3)^2}} \cdot \sqrt{b} = 3b^3\sqrt{b} \end{aligned} $$

Exercise 3: Simplify expression

Level 1

$$ \color{blue}{\sqrt{16a^5}} $$ $ 4a\sqrt a $
$ 8a\sqrt a $
$ 8a^2 $
$ 4a^2 \sqrt a $

Level 2

$$ \color{blue}{\sqrt{8x^3y^3}} $$ $ x\sqrt{2xy} $
$ 2xy\sqrt{2x} $
$ 2\sqrt 2\, x^2 y $
$ 2\sqrt 2\, x y^2 $