1. **State the problem:** Simplify the expression $$x^{\frac{2}{5}} \left(x^{\frac{6}{5}} - x^{\frac{7}{2}}\right)$$ and express the answer with positive exponents. 2. **Recall the exponent rules:** When multiplying terms with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$ 3. **Apply the distributive property:** Multiply $$x^{\frac{2}{5}}$$ by each term inside the parentheses: $$x^{\frac{2}{5}} \cdot x^{\frac{6}{5}} - x^{\frac{2}{5}} \cdot x^{\frac{7}{2}}$$ 4. **Add exponents for each product:** - For the first term: $$\frac{2}{5} + \frac{6}{5} = \frac{8}{5}$$ - For the second term: $$\frac{2}{5} + \frac{7}{2} = \frac{2}{5} + \frac{35}{10} = \frac{4}{10} + \frac{35}{10} = \frac{39}{10}$$ 5. **Rewrite the expression:** $$x^{\frac{8}{5}} - x^{\frac{39}{10}}$$ 6. **Final answer:** The simplified expression with positive exponents is $$x^{\frac{8}{5}} - x^{\frac{39}{10}}$$ --- **Note on the rectangle in the graph:** The rectangle with vertices at (1,3) and (7,5) has width $$7 - 1 = 6$$ and height $$5 - 3 = 2$$, but this is additional information and not needed for the algebraic simplification.