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Roots and Radicals: (lesson 2 of 3)

Adding and Subtracting Radical Expressions

The steps in adding and subtracting Radical are:

Step 1. Simplify radicals. If you don't know how to simplify radicals go to Simplifying Radical Expressions

Step 2. Combine like radicals.

Example 1: Add or subtract to simplify radical expression: 212+27 2 \sqrt{12} + \sqrt{27}

Solution:

Step 1: Simplify radicals

12=4⋅3=4⋅3=2327=9⋅3=9⋅3=33 \begin{aligned} \sqrt{12} &= \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}\\ \sqrt{27} &= \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3} \end{aligned}

Step 2: Combine like radicals

212+27=2⋅23+33=43+33=73⏟COMBINE LIKE TERMS 2 \color{red}{\sqrt{12}} + \color{blue}{\sqrt{27}} = 2\cdot \color{red}{2 \sqrt{3}} + \color{blue}{3\sqrt{3}} = \underbrace{ 4\sqrt{3} + 3\sqrt{3} = 7\sqrt{3}}_\text{COMBINE LIKE TERMS}

Example 2: Add or subtract to simplify radical expression: 350−28−532 3 \sqrt{50} - 2 \sqrt{8} - 5 \sqrt{32}

Solution:

Step 1: Simplify radicals

50=25⋅2=528=4⋅2=2232=16⋅2=42 \begin{aligned} \sqrt{50} &= \sqrt{25 \cdot 2} = 5 \sqrt{2} \\ \sqrt{8} &= \sqrt{4 \cdot 2} = 2 \sqrt{2} \\ \sqrt{32} &= \sqrt{16 \cdot 2} = 4 \sqrt{2} \end{aligned}

Step 2: Combine like radicals

350−28−532==3⋅52−2⋅22−5⋅42==152−42−202=−92⏟COMBINE LIKE TERMS \begin{aligned} 3 \color{red}{\sqrt{50}} - 2 \color{blue}{\sqrt{8}} - 5 \color{green}{\sqrt{32}} &= \\ &= 3 \cdot \color{red}{5 \sqrt{2}} - 2 \cdot \color{blue}{2 \sqrt{2}} - 5 \cdot \color{green}{4 \sqrt{2}} = \\ &= \underbrace{ 15 \sqrt{2} - 4 \sqrt{2} - 20 \sqrt{2} = -9 \sqrt{2}}_\text{COMBINE LIKE TERMS} \end{aligned}

Exercise 1: Add or subtract to simplify

Level 1

50−32= \color{blue}{\sqrt{50} - \sqrt{32} = } −3 -\sqrt{3}
−2+3 -\sqrt{2} + \sqrt{3}
−2 -\sqrt{2}
2 \sqrt{2}

Level 2

212−327 \color{blue}{2\sqrt{12} - 3 \sqrt{27}} −53 -5\sqrt{3}
33 3\sqrt{3}
42 4\sqrt{2}
−42 -4\sqrt{2}

Example 3: Add or subtract to simplify radical expression: 42−33 4 \sqrt{2} - 3 \sqrt{3}

Solution:

Here the radicands differ and are already simplified, so this expression cannot be simplified.

Adding and Subtracting Radicals with Fractions

Example 4: Add or subtract to simplify radical expression: 4209+54516 4 \sqrt{ \frac{20}{9} } + 5 \sqrt{ \frac{45}{16} }

Solution:

Step 1: Simplify radicals

209=209=4⋅53=2⋅53=23⋅54516=4516=9⋅54=3⋅54=34⋅5 \begin{aligned} \color{blue}{\sqrt{ \frac{20}{9} }} &= \frac{\sqrt{20}}{\sqrt{9}} = \frac{\sqrt{4 \cdot 5}}{3} = \frac{2 \cdot \sqrt{5}}{3} = \color{blue}{\frac{2}{3} \cdot \sqrt{5}} \\ \color{red}{\sqrt{ \frac{45}{16} }} &= \frac{\sqrt{45}}{\sqrt{16}} = \frac{\sqrt{9 \cdot 5}}{4} = \frac{3 \cdot \sqrt{5}}{4} = \color{red}{\frac{3}{4} \cdot \sqrt{5}} \\ \end{aligned}

Step 2: Combine like radicals

4⋅209+5⋅4516==4⋅23⋅5+5⋅34⋅5==83⋅5+154⋅5==(83+154)5=77125 \begin{aligned} 4 \cdot \color{blue}{\sqrt{\frac{20}{9}}} + 5 \cdot \color{red}{\sqrt{\frac{45}{16}}} &= \\ &= 4 \cdot \color{blue}{\frac{2}{3} \cdot \sqrt{5}} + 5 \cdot \color{red}{\frac{3}{4} \cdot \sqrt{5}} = \\ &= \frac{8}{3} \cdot \sqrt{5} + \frac{15}{4} \cdot \sqrt{5} = \\ &= \left( \frac{8}{3} + \frac{15}{4} \right) \sqrt{5} = \frac{77}{12} \sqrt{5} \end{aligned}

Example 5: Add or subtract to simplify radical expression: 624x4−354x4 6 \sqrt{ \frac{24}{x^4}} - 3 \sqrt{ \frac{54}{x^4}}

Solution:

Step 1: Simplify radicals

24x4=24x4=4⋅6x2=26x254x4=54x4=9⋅6x2=36x2 \begin{aligned} \color{blue}{\sqrt{\frac{24}{x^4}}} &= \frac{\sqrt{24}}{\sqrt{x^4}} = \frac{\sqrt{4 \cdot 6}}{x^2} = \color{blue}{\frac{2 \sqrt{6}}{x^2}} \\ \color{red}{\sqrt{\frac{54}{x^4}}} &= \frac{\sqrt{54}}{\sqrt{x^4}} = \frac{\sqrt{9 \cdot 6}}{x^2} = \color{red}{\frac{3 \sqrt{6}}{x^2}} \end{aligned}

Step 2: Combine like radicals

6⋅24x4−3⋅54x4=6⋅26x2−3⋅36x2=126x2−96x2=126−96x2=36x2 6 \cdot \color{blue}{\sqrt{\frac{24}{x^4}}} - 3 \cdot \color{red}{\sqrt{\frac{54}{x^4}}} = 6 \cdot \color{blue}{\frac{2 \sqrt{6}}{x^2}} - 3 \cdot \color{red}{\frac{3 \sqrt{6}}{x^2}} = \frac{12 \sqrt{6}}{x^2} - \frac{9 \sqrt{6}}{x^2} = \frac{12 \sqrt{6} - 9 \sqrt{6}}{x^2} = \frac{3\sqrt{6}}{x^2}

Exercise 2: Add or subtract to simplify radical expression

Level 1

434+82716 \color{blue}{4\sqrt{\frac{3}{4}} + 8 \sqrt{ \frac{27}{16}} } −22 -2\sqrt{2}
−32 -3\sqrt{2}
82 8\sqrt{2}
−42 -4\sqrt{2}

Level 2

33a2−212a2 \color{blue}{ 3\sqrt{\frac{3}{a^2}} - 2 \sqrt{\frac{12}{a^2}}} 3a \frac{\sqrt3}{a}
−3a -\frac{\sqrt3}{a}
−a3 -a\sqrt3
a3 a\sqrt3

Algebra

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Analytic geometry

Linear Algebra

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More help with radical expressions at mathportal.org
Rationalize Radical Denominator - online calculator
Simplifying Radical Expressions - online calculator
Roots and radicals formulas
Multiplying and Dividing Radical Expressions - next lesson
Simplifying Radical Expressions - previous lesson