Algebra 1 Practice - Rationalize the Denominator (Example 1)

Please subscribe!    / nickperich   *Algebra 1 practice on rationalizing the denominator* involves rewriting fractions so that the denominator no longer contains any square roots (or other radicals). This process ensures that expressions are in a standard, simplified form. Steps to Rationalize the Denominator 1. *Identify the radical in the denominator.* If the denominator contains a square root, you'll multiply both numerator and denominator by a value that eliminates the square root. 2. *Multiply by the conjugate* (if necessary). When the denominator has two terms (e.g., \( a + \sqrt{b} \)), multiply by the conjugate, \( a - \sqrt{b} \), to remove the square root using the difference of squares. 3. *Simplify the expression.* After multiplying, simplify the numerator and denominator by combining like terms and reducing the fraction if possible. --- Examples #### Example 1: Single Radical in Denominator \[ \frac{5}{\sqrt{3}} \] *Solution:* Multiply numerator and denominator by \( \sqrt{3} \): \[ \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3} \] Final answer: \[ \frac{5\sqrt{3}}{3} \] --- #### Example 2: Denominator with a Binomial \[ \frac{2}{1 + \sqrt{5}} \] *Solution:* Multiply numerator and denominator by the conjugate \( 1 - \sqrt{5} \): \[ \frac{2}{1 + \sqrt{5}} \cdot \frac{1 - \sqrt{5}}{1 - \sqrt{5}} = \frac{2(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})} \] Use the difference of squares in the denominator: \[ \frac{2(1 - \sqrt{5})}{1 - 5} = \frac{2(1 - \sqrt{5})}{-4} \] Simplify: \[ \frac{2(1 - \sqrt{5})}{-4} = \frac{1 - \sqrt{5}}{-2} \] Final answer: \[ \frac{-1 + \sqrt{5}}{2} \] --- #### Example 3: Fraction with Radical Numerator and Denominator \[ \frac{\sqrt{2}}{\sqrt{6}} \] *Solution:* Multiply numerator and denominator by \( \sqrt{6} \): \[ \frac{\sqrt{2}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{12}}{6} \] Simplify \( \sqrt{12} \): \[ \frac{\sqrt{4 \cdot 3}}{6} = \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3} \] Final answer: \[ \frac{\sqrt{3}}{3} \] --- Practice Problems 1. \(\frac{3}{\sqrt{7}}\) 2. \(\frac{4}{2 + \sqrt{3}}\) 3. \(\frac{\sqrt{5}}{\sqrt{15}}\) 4. \(\frac{7}{3 - \sqrt{2}}\) 5. \(\frac{\sqrt{6}}{1 - \sqrt{2}}\) --- Key Skills Practiced Removing square roots from denominators. Using the difference of squares to simplify. Combining like terms and reducing fractions. This practice strengthens algebraic manipulation and prepares students for advanced work with radicals and rational expressions. I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths Please subscribe!    / nickperich   I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study