Simplify √25 · ∛(√25): Multiple Radical Expression Problem

To solve the problem $\sqrt{25} \cdot \sqrt[3]{\sqrt{25}}$, follow these steps:

• First, express each root in terms of exponents:
• $\sqrt{25} = 25^{1/2}$
• $\sqrt[3]{\sqrt{25}} = \sqrt[3]{25^{1/2}}$
• Using the law of exponents $(a^m)^n = a^{m \cdot n}$, simplify $\sqrt[3]{25^{1/2}}$ as follows:
• $(25^{1/2})^{1/3} = 25^{(1/2) \cdot (1/3)} = 25^{1/6}$
• Now, multiply the two expressions:
• $25^{1/2} \cdot 25^{1/6} = 25^{(1/2 + 1/6)}$
• Calculate the sum of the exponents: $\frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$
• This gives: $25^{2/3}$
• Recognize $25 = 5^2$, thus: $(5^2)^{2/3} = 5^{2 \cdot (2/3)} = 5^{4/3}$
• Convert mixed fraction: $5^{4/3} = 5^{1 + 1/3} = 5^{1\frac{1}{3}}$

Therefore, the product $\sqrt{25} \cdot \sqrt[3]{\sqrt{25}}$ equals $\mathbf{5^{1\frac{1}{3}}}$.