Simplify expressions involving rational exponents I 2 VX nplify. Assume all variables are positive.

Certainly! Simplifying expressions with rational (fractional) exponents is a fundamental concept in algebra. Rational exponents allow us to express roots and powers in a unified way. Here's a comprehensive guide to understanding and simplifying such expressions. ### **Understanding Rational Exponents** A **rational exponent** is an exponent that is a fraction, such as \( \frac{m}{n} \). The general form is: \[ x^{\frac{m}{n}} \] This can be interpreted in two equivalent ways: 1. **Root Form**: \[ x^{\frac{m}{n}} = \sqrt[n]{x^m} \] This means you take the \( n \)-th root of \( x \) raised to the \( m \)-th power. 2. **Power Form**: \[ x^{\frac{m}{n}} = \left( \sqrt[n]{x} \right)^m \] This means you first take the \( n \)-th root of \( x \) and then raise the result to the \( m \)-th power. ### **Key Properties of Exponents** To simplify expressions with rational exponents, it's essential to recall some exponent rules: 1. **Product of Powers**: \[ x^a \cdot x^b = x^{a + b} \] 2. **Quotient of Powers**: \[ \frac{x^a}{x^b} = x^{a - b} \] 3. **Power of a Power**: \[ \left( x^a \right)^b = x^{a \cdot b} \] 4. **Power of a Product**: \[ (xy)^a = x^a \cdot y^a \] 5. **Power of a Quotient**: \[ \left( \frac{x}{y} \right)^a = \frac{x^a}{y^a} \] ### **Simplification Steps** Let's go through a step-by-step example to illustrate how to simplify an expression with rational exponents. **Example: Simplify \( \frac{x^{\frac{3}{4}} \cdot x^{\frac{2}{3}}}{x^{\frac{1}{2}}} \)** **Step 1: Apply the Product of Powers Rule** When multiplying like bases, add the exponents: \[ x^{\frac{3}{4}} \cdot x^{\frac{2}{3}} = x^{\frac{3}{4} + \frac{2}{3}} \] To add the exponents, find a common denominator (12 in this case): \[ \frac{3}{4} = \frac{9}{12}, \quad \frac{2}{3} = \frac{8}{12} \] \[ \frac{9}{12} + \frac{8}{12} = \frac{17}{12} \] \[ x^{\frac{3}{4}} \cdot x^{\frac{2}{3}} = x^{\frac{17}{12}} \] **Step 2: Apply the Quotient of Powers Rule** Now, divide by \( x^{\frac{1}{2}} \): \[ \frac{x^{\frac{17}{12}}}{x^{\frac{1}{2}}} = x^{\frac{17}{12} - \frac{1}{2}} \] Convert \( \frac{1}{2} \) to twelfths: \[ \frac{1}{2} = \frac{6}{12} \] \[ \frac{17}{12} - \frac{6}{12} = \frac{11}{12} \] \[ \frac{x^{\frac{17}{12}}}{x^{\frac{1}{2}}} = x^{\frac{11}{12}} \] **Final Simplified Form:** \[ x^{\frac{11}{12}} \] ### **Another Example with Variables** **Example: Simplify \( \left( \frac{a^{\frac{2}{3}} \cdot b^{\frac{3}{4}}}{a^{\frac{1}{6}}} \right)^2 \)** **Step 1: Simplify Inside the Parentheses** Apply the product and quotient rules for exponents: \[ \frac{a^{\frac{2}{3}}}{a^{\frac{1}{6}}} = a^{\frac{2}{3} - \frac{1}{6}} = a^{\frac{4}{6} - \frac{1}{6}} = a^{\frac{3}{6}} = a^{\frac{1}{2}} \] \[ \text{Thus, the expression inside becomes } a^{\frac{1}{2}} \cdot b^{\frac{3}{4}} \] **Step 2: Apply the Power Outside the Parentheses** Raise each part to the power of 2: \[ \left( a^{\frac{1}{2}} \cdot b^{\frac{3}{4}} \right)^2 = \left( a^{\frac{1}{2}} \right)^2 \cdot \left( b^{\frac{3}{4}} \right)^2 = a^{1} \cdot b^{\frac{6}{4}} = a \cdot b^{\frac{3}{2}} \] **Final Simplified Form:** \[ a \cdot b^{\frac{3}{2}} \quad \text{or} \quad a \sqrt{b^3} = a b^{1.5} \] ### **Converting Between Radical and Exponent Form** Sometimes, it's helpful to express the final answer in radical form, especially if it simplifies the expression: - **Example 1: \( x^{\frac{3}{2}} \)** \[ x^{\frac{3}{2}} = \left( \sqrt{x} \right)^3 = \sqrt{x^3} = x \sqrt{x} \] - **Example 2: \( y^{-\frac{4}{5}} \)** \[ y^{-\frac{4}{5}} = \frac{1}{y^{\frac{4}{5}}} = \frac{1}{\sqrt[5]{y^4}} \] ### **Summary of Steps to Simplify Expressions with Rational Exponents** 1. **Identify like bases** and apply the product or quotient rules to combine exponents. 2. **Convert negative exponents** to positive by moving the term to the numerator or denominator. 3. **Simplify the exponents** by performing the arithmetic on the fractions. 4. **Express in radical form** if it makes the expression clearer or simpler. 5. **Factor expressions** inside radicals when possible to simplify further. ### **Practice Problem** **Simplify:** \( \frac{m^{\frac{5}{6}} \cdot m^{\frac{1}{3}}}{m^{\frac{1}{2}}} \) **Solution:** 1. **Combine exponents in the numerator:** \[ m^{\frac{5}{6}} \cdot m^{\frac{1}{3}} = m^{\frac{5}{6} + \frac{2}{6}} = m^{\frac{7}{6}} \] 2. **Divide by \( m^{\frac{1}{2}} \):** \[ \frac{m^{\frac{7}{6}}}{m^{\frac{1}{2}}} = m^{\frac{7}{6} - \frac{3}{6}} = m^{\frac{4}{6}} = m^{\frac{2}{3}} \] 3. **Optional - Express in radical form:** \[ m^{\frac{2}{3}} = \sqrt[3]{m^2} \] **Final Answer:** \( m^{\frac{2}{3}} \) or \( \sqrt[3]{m^2} \) --- Feel free to provide a specific expression you'd like to simplify if you need further assistance!