Rationalize Denominators - Questions with Solutions

Grade 10 questions on how to rationalize radical expressions with solutions are presented.

Examples with Solutions

1. \(\quad \sqrt{x} \cdot \sqrt{x} = (\sqrt{x})^2 = x\)
2. \(\quad \sqrt[3]{x} \cdot (\sqrt[3]{x})^2 = (\sqrt[3]{x})^3 = x\)
3. \(\quad (\sqrt{x} - \sqrt{y}) (\sqrt{x} + \sqrt{y}) = (\sqrt{x})^2 - (\sqrt{y})^2 = x - y\)
4. \(\quad (x - \sqrt{y}) (x + \sqrt{y}) = x^2 - (\sqrt{y})^2 = x^2 - y\)

Examples

1. 
   \[ \dfrac{1}{\sqrt{2} } \]

   solution

   Because of \( \sqrt 2 \) in the denominator, multiply numerator and denominator by \( \sqrt 2 \) and simplify \[ \dfrac{1}{\sqrt{2}} = \dfrac{1}{\sqrt{2}} \cdot \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{\sqrt{2}}{(\sqrt{2})^2} = \dfrac{\sqrt{2}}{2} \]
2. 
   \[ \dfrac{1}{\sqrt[3]{x}} \]

   solution

   Because of \( \sqrt[3]{x} \) in the denominator, multiply the numerator and denominator by \( \left( \sqrt[3]{x} \right)^2 \) and simplify. \[ \dfrac{1}{\sqrt[3]{x}} = \dfrac{1}{\sqrt[3]{x}} \cdot \dfrac{\left( \sqrt[3]{x} \right)^2}{\left( \sqrt[3]{x} \right)^2} = \dfrac{\sqrt[3]{x^2}}{x} \]
3. 
   \[ \dfrac{4}{\sqrt{3} - \sqrt{2}} \]

   solution

   Because of the expression \( \sqrt{3} - \sqrt{2} \) in the denominator, multiply the numerator and denominator by the conjugate of \( \sqrt{3} - \sqrt{2} \) which is \( \sqrt{3} + \sqrt{2} \) to obtain: \[ \dfrac{4}{\sqrt{3} - \sqrt{2}} = \dfrac{4}{\sqrt{3} - \sqrt{2}} \cdot \dfrac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} \] \[ = \dfrac{4(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \] \[ = \dfrac{4(\sqrt{3} + \sqrt{2})}{(\sqrt{3})^{2} - (\sqrt{2})^{2}} \] \[ = \dfrac{4(\sqrt{3} + \sqrt{2})}{3 - 2} = 4(\sqrt{3} + \sqrt{2}) \]
4. 
   \[ \dfrac{5x^{2}}{\sqrt[3]{x^{2}}} \]

   solution

   Because of the expression \( \sqrt[3]{x^2} \) in the denominator, multiply the numerator and denominator by \( \left( \sqrt[3]{x^2} \right)^2 \) to obtain \[ \dfrac{5x^2}{\sqrt[3]{x^2}} = \dfrac{5x^2}{\sqrt[3]{x^2}} \cdot \dfrac{\left( \sqrt[3]{x^2} \right)^2}{\left( \sqrt[3]{x^2} \right)^2} \] \[ = \dfrac{5x^2 \sqrt[3]{x^4}}{\left( \sqrt[3]{x^2} \right)^3} \] Simplify and cancel terms: \[ = \dfrac{5x^2 \sqrt[3]{x^4}}{x^2} = 5\sqrt[3]{x^4} = 5x\sqrt[3]{x} \]
5. 
   \[ 5) \quad \dfrac{x^2}{y + \sqrt{x^2 + y^2}} \]

   solution

   Because of the expression \( y + \sqrt{x^2 + y^2} \) in the denominator, multiply numerator and denominator by its conjugate \( y - \sqrt{x^2 + y^2} \) to obtain \[ \dfrac{x^2}{y + \sqrt{x^2 + y^2}} = \dfrac{x^2}{y + \sqrt{x^2 + y^2}} \cdot \dfrac{y - \sqrt{x^2 + y^2}}{y - \sqrt{x^2 + y^2}} \] \[ = \dfrac{x^2(y - \sqrt{x^2 + y^2})}{(y)^2 - (\sqrt{x^2 + y^2})^2} \] \[ = \dfrac{x^2(y - \sqrt{x^2 + y^2})}{y^2 - (x^2 + y^2)} \] \[ = \dfrac{x^2(y - \sqrt{x^2 + y^2})}{-x^2} \] \[ = -y + \sqrt{x^2 + y^2} \]

Questions

1. \( \quad \dfrac{10}{\sqrt{5}} \)
2. \( \quad 2\sqrt{2}\sqrt{3} - \dfrac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \)
3. \( \quad \dfrac{7x^4}{\sqrt[3]{x^4}} \)
4. \( \quad \dfrac{-x^2}{y + \sqrt{x^2 + y^2}} \)

Solutions to the Above Problems

1. 
   Multiply numerator and denominator by \( \sqrt 5 \) \[ \frac{10}{\sqrt{5}} = \frac{10}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{(\sqrt{5})^2} = \frac{10\sqrt{5}}{5} = 2\sqrt{5} \]
2. 
   Multiply numerator and denominator by \( \sqrt{2} - \sqrt{3} \) \[ 2\sqrt{2}\sqrt{3} - \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} = 2\sqrt{2}\sqrt{3} - \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \cdot \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} \] and simplify \[ = 2\sqrt{2}\sqrt{3} - \frac{(\sqrt{2}-\sqrt{3})^2}{(\sqrt{2})^2-(\sqrt{3})^2} \] \[ = 2\sqrt{2}\sqrt{3} - \frac{(\sqrt{2})^2 + (\sqrt{3})^2 - 2\sqrt{2}\sqrt{3}}{2-3} \] \[ = 2\sqrt{2}\sqrt{3} - \frac{2+3-2\sqrt{2}\sqrt{3}}{-1} \] \[ = 2\sqrt{2}\sqrt{3} + 5 - 2\sqrt{2}\sqrt{3} \] \[ = 5 \]
3. 
   Multiply numerator and denominator by \( \left( \sqrt[3]{x^4} \right)^2 \) \[ \frac{7x^4}{\sqrt[3]{x^4}} = \frac{7x^4}{\sqrt[3]{x^4}} \cdot \frac{\left( \sqrt[3]{x^4} \right)^2}{\left( \sqrt[3]{x^4} \right)^2} \] and simplify \[ = \frac{7x^4 \sqrt[3]{x^8}}{\left( \sqrt[3]{x^4} \right)^3} = \frac{7x^4 \sqrt[3]{x^8}}{x^4} = 7 \sqrt[3]{x^8} = 7x^2 \sqrt[3]{x^2} \]
4. 
   Multiply numerator and denominator by \( y - \sqrt{x^2 + y^2} \) \[ \frac{-x^2}{y + \sqrt{x^2 + y^2}} = \frac{-x^2}{y + \sqrt{x^2 + y^2}} \cdot \frac{y - \sqrt{x^2 + y^2}}{y - \sqrt{x^2 + y^2}} \] and simplify \[ = \frac{-x^2(y - \sqrt{x^2 + y^2})}{y^2 - (x^2 + y^2)} \] \[ = \frac{-x^2(y - \sqrt{x^2 + y^2})}{-x^2} = y - \sqrt{x^2 + y^2} \]

More References and links

• Simplify Radical Expressions - Questions with Solutions for Grade 10
• High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers
• Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
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