Properties of Rational Exponents

4. Problem-Solving Techniques

• Prioritize Unifying Forms: When encountering problems that contain both radicals and rational exponents, prioritize converting the radicals to the form of rational exponents (i.e.,  $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ ), and then use the properties of rational exponents for calculation to avoid the complexity of radical operations. For example, to simplify  $\sqrt{2} \cdot \sqrt[3]{2}$ , it can be converted to  $2^{\frac{1}{2}} \cdot 2^{\frac{1}{3}}$ , and then calculated using the "Multiplication of Powers with the Same Base" property.
• Clarify the Order of Exponent Operations: Follow the order of "first calculate the inside of the exponent, then calculate the power, and finally calculate multiplication and division". If there are parentheses, calculate the content inside the parentheses first. For example, when calculating  $(a^{\frac{1}{2}} \cdot a^{\frac{1}{3}})^4$ , first calculate the expression inside the parentheses:  $a^{\frac{1}{2}+\frac{1}{3}} = a^{\frac{5}{6}}$ , and then calculate  $(a^{\frac{5}{6}})^4 = a^{\frac{20}{6}} = a^{\frac{10}{3}}$ .
• Skillfully Use Inverse Operations of Properties: In addition to applying the properties in the forward direction, it is also necessary to flexibly use their inverse operations, such as the inverse operation of the "Power of a Product" property ( $a^m \cdot b^m = (a \cdot b)^m$ ) and the inverse operation of the "Power of a Power" property ( $a^{m \cdot n} = (a^m)^n = (a^n)^m$ ). For example, to simplify  $2^{\frac{2}{3}} \cdot 4^{\frac{2}{3}}$ , the inverse operation of the "Power of a Product" property can be used to get  $(2 \times 4)^{\frac{2}{3}} = 8^{\frac{2}{3}} = 4$ .
• Techniques for Handling Negative Exponents: Strictly follow the "Negative Exponent" property  $a^{-m} = \frac{1}{a^m}$  ( $a\neq0$ ), convert negative exponents to the reciprocals of positive exponents before performing operations, and avoid sign errors. For example, when calculating  $5^{-\frac{1}{2}} \cdot 5^{\frac{3}{2}}$ , first convert it to  $\frac{1}{5^{\frac{1}{2}}} \cdot 5^{\frac{3}{2}}$ , and then use the "Multiplication of Powers with the Same Base" property to get  $5^{\frac{3}{2} - \frac{1}{2}} = 5^1 = 5$ .
• Unify Bases to Simplify Operations: When the bases in the expression are different but can be converted to the same base (such as  $4=2^2$ ,  $8=2^3$ ,  $9=3^2$ , etc.), first unify the bases, and then use the properties of powers with the same base for calculation. For example, to simplify  $\frac{4^{\frac{1}{2}}}{2^{\frac{1}{3}}}$ , convert 4 to  $2^2$ , so  $4^{\frac{1}{2}} = (2^2)^{\frac{1}{2}} = 2^1$ , and the expression becomes  $\frac{2^1}{2^{\frac{1}{3}}} = 2^{1 - \frac{1}{3}} = 2^{\frac{2}{3}}$ .