1. **State the problem:** Rewrite the expression $$8^{0.5} \cdot 2^{\frac{3}{5}}$$ as a single power of 2. 2. **Recall the formula and rules:** - Any number can be expressed as a power of its prime factors. - The product of powers with the same base can be combined by adding exponents: $$a^m \cdot a^n = a^{m+n}$$ 3. **Express 8 as a power of 2:** Since $$8 = 2^3$$, rewrite the expression: $$8^{0.5} = (2^3)^{0.5}$$ 4. **Apply the power of a power rule:** $$(a^m)^n = a^{m \cdot n}$$ So, $$(2^3)^{0.5} = 2^{3 \times 0.5} = 2^{1.5}$$ 5. **Rewrite the original expression:** $$8^{0.5} \cdot 2^{\frac{3}{5}} = 2^{1.5} \cdot 2^{\frac{3}{5}}$$ 6. **Add the exponents since bases are the same:** $$2^{1.5} \cdot 2^{\frac{3}{5}} = 2^{1.5 + \frac{3}{5}}$$ 7. **Convert decimals to fractions for addition:** $$1.5 = \frac{3}{2}$$ So, $$\frac{3}{2} + \frac{3}{5} = \frac{3 \times 5}{2 \times 5} + \frac{3 \times 2}{5 \times 2} = \frac{15}{10} + \frac{6}{10} = \frac{21}{10}$$ 8. **Final expression:** $$2^{\frac{21}{10}}$$ **Answer:** The expression $$8^{0.5} \cdot 2^{\frac{3}{5}}$$ rewritten as a single power of 2 is $$2^{\frac{21}{10}}$$. This process is called **exponentiation and simplification of powers**.