Simplify each expression. 19. \( 81^{\frac{1}{2}} \) 20. \( 34^{\frac{1}{3}} \) 21. \( 64^{\frac{2}{3}} \) 22. \( \left(2^{6}\right)^{\frac{1}{2}} \)

The square root of 81 is equal to 9, so \( 81^{\frac{1}{2}} = 9 \). Perfect squares are like treasure chests; when you find one, it unlocks a number that you may not have realized was hiding inside! For \( 34^{\frac{1}{3}} \), we are looking for the cube root of 34, which is approximately 3.3. Cube roots are the cool cousins of square roots, often making us rethink how we view numbers by expanding our understanding beyond plain integer roots! Next, \( 64^{\frac{2}{3}} \) can be computed by first finding the cube root of 64, which is 4, and then squaring that result, giving us \( 4^2 = 16 \). Think of cube roots as a science project: you break it down to its simplest form before soaring to higher dimensions like a rocket! Lastly, for \( \left(2^{6}\right)^{\frac{1}{2}} \), you multiply the exponents: \( 2^{6 \cdot \frac{1}{2}} = 2^{3} \), which equals 8. It’s like a fun game of stacking: you combine and simplify to see how high your numbers can reach!