1. The problem is to understand the properties of rational exponents and how they relate to integral exponents. 2. The key formula is that for positive real numbers $a$ and $b$, and rational numbers $m$ and $n$, the properties of exponents hold: - Product of powers: $$a^m \cdot a^n = a^{m+n}$$ - Power of a power: $$(a^m)^n = a^{mn}$$ - Power of a product: $$(ab)^m = a^m b^m$$ 3. These properties are the same as those for integral exponents but now apply when $m$ and $n$ are rational numbers (fractions). 4. For example, if $m=\frac{1}{2}$ and $n=\frac{1}{3}$, then: $$a^{\frac{1}{2}} \cdot a^{\frac{1}{3}} = a^{\frac{1}{2} + \frac{1}{3}} = a^{\frac{3}{6} + \frac{2}{6}} = a^{\frac{5}{6}}$$ 5. This shows how to add rational exponents when multiplying powers with the same base. 6. Another example for power of a power: $$(a^{\frac{1}{2}})^{3} = a^{\frac{1}{2} \cdot 3} = a^{\frac{3}{2}}$$ 7. These properties allow us to manipulate expressions with roots and fractional powers easily, extending the rules from integers to rational numbers. Final answer: The properties of integral exponents extend naturally to rational exponents, allowing the same algebraic manipulations with fractional powers.