Algebra 2 Practice - Rewrite a Fraction Exponent in Radical Form (Example 1)

Please subscribe!    / nickperich   *Algebra 2 Practice - Rewrite a Fraction Exponent in Radical Form* Fraction exponents are another way of expressing roots and powers. The key is understanding the relationship between powers and roots. A fraction exponent can be rewritten as a radical expression using the following rule: \[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \] This means that a base \(a\) raised to the power of \( \frac{m}{n} \) can be rewritten as the *n-th root* of \(a^m\). #### **Steps to Rewrite a Fraction Exponent in Radical Form**: 1. *Identify the denominator* of the fraction in the exponent. This will tell you the root (i.e., square root, cube root, etc.). 2. *Identify the numerator* of the fraction. This tells you the power to which the base is raised. 3. *Apply the root and power* as specified by the fraction exponent. --- *Example 1:* Rewrite \( x^{\frac{3}{2}} \) in radical form. 1. The denominator of the fraction is 2, so this indicates a **square root**. 2. The numerator is 3, so this indicates the base \(x\) is raised to the **third power**. Therefore, \( x^{\frac{3}{2}} \) can be written as: \[ x^{\frac{3}{2}} = \sqrt{x^3} \] --- *Example 2:* Rewrite \( y^{\frac{4}{3}} \) in radical form. 1. The denominator of the fraction is 3, so this indicates a **cube root**. 2. The numerator is 4, so this indicates the base \(y\) is raised to the **fourth power**. Therefore, \( y^{\frac{4}{3}} \) can be written as: \[ y^{\frac{4}{3}} = \sqrt[3]{y^4} \] --- *Example 3:* Rewrite \( a^{\frac{5}{4}} \) in radical form. 1. The denominator of the fraction is 4, so this indicates a **fourth root**. 2. The numerator is 5, so this indicates the base \(a\) is raised to the **fifth power**. Therefore, \( a^{\frac{5}{4}} \) can be written as: \[ a^{\frac{5}{4}} = \sqrt[4]{a^5} \] --- *Example 4:* Rewrite \( 8^{\frac{2}{3}} \) in radical form. 1. The denominator of the fraction is 3, so this indicates a **cube root**. 2. The numerator is 2, so this indicates the base 8 is raised to the **second power**. Therefore, \( 8^{\frac{2}{3}} \) can be written as: \[ 8^{\frac{2}{3}} = \sqrt[3]{8^2} \] You can simplify this further since \( 8^2 = 64 \), so: \[ 8^{\frac{2}{3}} = \sqrt[3]{64} \] And \( \sqrt[3]{64} = 4 \), so the final result is: \[ 8^{\frac{2}{3}} = 4 \] --- **Summary**: *Fraction exponents* can be rewritten as radicals by recognizing the denominator as the root and the numerator as the power. **Formula**: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \) The denominator tells you the root, and the numerator tells you the power. 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