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

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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!
To simplify expressions with rational exponents, follow these steps: 1. **Combine like bases** using the product and quotient rules. 2. **Simplify the exponents** by performing arithmetic on the fractions. 3. **Express in radical form** if it helps simplify the expression. 4. **Factor inside radicals** when possible. **Example:** Simplify $$ \frac{m^{\frac{5}{6}} \cdot m^{\frac{1}{3}}}{m^{\frac{1}{2}}} $$. - Combine exponents: $$ m^{\frac{5}{6} + \frac{2}{6}} = m^{\frac{7}{6}} $$ - Divide by $$ m^{\frac{1}{2}} $$: $$ m^{\frac{7}{6} - \frac{3}{6}} = m^{\frac{4}{6}} = m^{\frac{2}{3}} $$ - Optional radical form: $$ \sqrt[3]{m^2} $$ **Final Answer:** $$ m^{\frac{2}{3}} $$ or $$ \sqrt[3]{m^2} $$

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

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