Adding and Subtracting Radical Expressions

 

In order to add or subtract radical expressions, the indices of the roots must be equal and the radicands (the expressions inside the roots) must be identical.

$$\Large x \sqrt[n]{a } + y \sqrt[n]{a } -z \sqrt[n]{a } = (x+y-z) \sqrt[n]{a } $$ as in

 

Examples:

 

\( \bullet \quad \sqrt{ 3} + \sqrt{2 } \) (the radicands are different)

\( \bullet \quad \sqrt[3]{ 7} + \sqrt{7 } \) (the indices of the roots are different)

\( \bullet \quad 3\sqrt{ 5} + \sqrt{5 }-2 \sqrt{ 5} = (3+1-2 )\sqrt{ 5} = 2 \sqrt{5 } \)

 

Question 8

 

What is the result of the operation ?  \[ \sqrt{48 } + \sqrt{ 12} – \sqrt{ \frac{27}{4} } \]

\[ \text{A)} 2\sqrt{ 3} \quad \text{B) } \frac{5}{2}\sqrt{ 3} \quad \text{C) } 3\sqrt{ 3} \quad \text{D) } 4\sqrt{ 3} \quad \text{E)} \frac{9}{2}\sqrt{ 3} \]

 

Solution:

 

\[ \sqrt{48} + \sqrt{12} – \sqrt{\frac{27}{4}} = \sqrt{3 \cdot 4^2} + \sqrt{3 \cdot 2^2} – \sqrt{\frac{3 \cdot 3^2}{2^2}} \] \[ = 4\sqrt{3} + 2\sqrt{3} – \frac{3\sqrt{3}}{2} \] \[ = \left(4 + 2 – \frac{3}{2}\right)\sqrt{3} = \frac{9}{2}\sqrt{3} \]

\(\textbf{Answer: E} \)

 

Question 9

 

What is the result of the operation?  \[ \sqrt[3]{ 128} + \sqrt[3]{ 16} -\sqrt[3]{ 250} \]

\[ \text{A)} -2 \sqrt[3]{ 2} \quad \text{B) } -\sqrt[3]{ 2} \quad \text{C) } 0 \quad \text{D) } \sqrt[3]{ 2} \quad \text{E)} 2 \sqrt[3]{ 2} \]

 

Solution:

 

\sqrt[3]{128} + \sqrt[3]{16} - \sqrt[3]{250} = \sqrt[3]{2 \cdot 4^3} + \sqrt[3]{2 \cdot 2^3} - \sqrt[3]{2 \cdot 5^3} = 4\sqrt[3]{2} + 2\sqrt[3]{2} - 5\sqrt[3]{2} = (4 + 2 - 5)\sqrt[3]{2} = \sqrt[3]{2}.

\(\textbf{Answer: D} \)

 

Question 10

 

What is the result of the operation ?  \[ \sqrt[9]{ 8} + \sqrt[3]{ -128} -\sqrt[12]{ 16} \]

\[ \text{A)} -2 \sqrt[3]{ 2} \quad \text{B) } -\sqrt[3]{ 2} \quad \text{C) } 0 \quad \text{D) } \sqrt[2]{ 2} \quad \text{E)} 2 \sqrt[2]{ 2} \]

 

Solution:

 

\[ =\sqrt[9]{ 8} + \sqrt[3]{ -128} -\sqrt[12]{ 16} \]

\[ =\sqrt[3.3]{ 2^3} + \sqrt[3]{ 2 \cdot (-4)^3}+ \sqrt[3.4]{ 2^4} \]

\[= \sqrt[3]{ 2} -4 \sqrt[3]{ 2 }+ \sqrt[3]{ 2} \]

\[= (1-4+1) \sqrt[3]{2 } = -2 \sqrt[3]{ 2} \]

 

\(\textbf{Answer: A} \)

 

 

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