Simplify WITHOUT using a calculator: \( 2.1 .1 \quad(\sqrt{2}+\sqrt{3})(2 \sqrt{3}-2 \sqrt{2}) \) \( 2.1 .2 \quad \sqrt[y]{\frac{6^{y}+5\left(3^{y}\right)}{10^{y}+5^{y+1}}} \)

To simplify \( (\sqrt{2}+\sqrt{3})(2 \sqrt{3}-2 \sqrt{2}) \), we can first factor out the 2 from the second term, giving us: \[ 2(\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2}). \] Now, using the distributive property (FOIL), we find: \[ 2[(\sqrt{2} \cdot \sqrt{3}) + (\sqrt{3} \cdot \sqrt{3}) - (\sqrt{2} \cdot \sqrt{2}) - (\sqrt{2} \cdot \sqrt{3})]. \] This simplifies to: \[ 2[3 - 2] = 2. \] Thus, the simplified expression is \( 2 \). Now for the second expression \( \sqrt[y]{\frac{6^{y}+5\left(3^{y}\right)}{10^{y}+5^{y+1}}} \): Starting with the numerator \( 6^y + 5(3^y) \), we can write \( 6^y \) as \( (2 \cdot 3)^y = 2^y \cdot 3^y \). Thus, the numerator becomes \( 2^y \cdot 3^y + 5 \cdot 3^y = (2^y + 5) \cdot 3^y \). For the denominator, we have \( 10^y + 5^{y+1} \). Write \( 10^y \) as \( (2 \cdot 5)^y = 2^y \cdot 5^y \): \[ 10^y + 5^{y+1} = 2^y \cdot 5^y + 5^{y+1} = 2^y \cdot 5^y + 5 \cdot 5^y = (2^y + 5) \cdot 5^y. \] Putting it all together gives us: \[ \sqrt[y]{\frac{(2^y + 5) \cdot 3^y}{(2^y + 5) \cdot 5^y}} = \sqrt[y]{\frac{3^y}{5^y}} = \sqrt[y]{\left(\frac{3}{5}\right)^y} = \frac{3}{5}. \] Thus, the simplified result is \( \frac{3}{5} \).