Math Calculators, Lessons and Formulas

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Roots and Radicals: (lesson 1 of 3)

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

 $$a) \sqrt{\color{red}{6}} \cdot \sqrt{\color{blue}{5}} = \sqrt{\color{red}{6} \cdot \color{blue}{5}} = \sqrt{30}$$ $$b) \sqrt{\color{red}{5}} \cdot \sqrt{\color{blue}{2ab}} = \sqrt{\color{red}{5} \cdot \color{blue}{2ab}} = \sqrt{10ab}$$ $$c) \sqrt[4]{\color{red}{4a}} \cdot \sqrt[4]{\color{blue}{7a^2b}} = \sqrt[4]{\color{red}{4a} \cdot \color{blue}{7a^2b}} = \sqrt[4]{28a^3b}$$

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

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

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

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

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 b^7} = \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

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