4) \( 55^{\frac{4}{3}} \) Simplify \( 2^{\frac{2}{3}} \cdot 4^{\frac{2}{3}} \)

Let's simplify each expression step by step. ### 1. Simplifying \( 55^{\frac{4}{3}} \) The expression \( 55^{\frac{4}{3}} \) can be rewritten using the properties of exponents: \[ 55^{\frac{4}{3}} = \left(55^{\frac{1}{3}}\right)^4 = 55 \cdot \left(55^{\frac{1}{3}}\right) \] Alternatively, it can be expressed in radical form: \[ 55^{\frac{4}{3}} = 55 \cdot \sqrt[3]{55} \] Since 55 is a product of prime numbers (55 = 5 × 11) and does not contain any perfect cubes, this is the simplest form. **Simplified Form:** \[ 55^{\frac{4}{3}} = 55 \cdot \sqrt[3]{55} \] ### 2. Simplifying \( 2^{\frac{2}{3}} \cdot 4^{\frac{2}{3}} \) First, express 4 as a power of 2: \[ 4 = 2^2 \] Now, substitute and simplify: \[ 2^{\frac{2}{3}} \cdot 4^{\frac{2}{3}} = 2^{\frac{2}{3}} \cdot (2^2)^{\frac{2}{3}} = 2^{\frac{2}{3}} \cdot 2^{\frac{4}{3}} = 2^{\frac{2}{3} + \frac{4}{3}} = 2^2 = 4 \] **Simplified Form:** \[ 2^{\frac{2}{3}} \cdot 4^{\frac{2}{3}} = 4 \] --- **Final Answers:** 1. \( 55^{\frac{4}{3}} = 55 \cdot \sqrt[3]{55} \) 2. \( 2^{\frac{2}{3}} \cdot 4^{\frac{2}{3}} = 4 \)