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

**Solution:** **2.1.1** Simplify \[ (\sqrt{2}+\sqrt{3})(2\sqrt{3}-2\sqrt{2}). \] 1. Factor out the common factor \(2\) from the second parenthesis: \[ (\sqrt{2}+\sqrt{3})(2\sqrt{3}-2\sqrt{2}) = 2 (\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2}). \] 2. Expand the expression in the parentheses: \[ (\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2}) = \sqrt{2}\sqrt{3} - (\sqrt{2})^2 + (\sqrt{3})^2 - \sqrt{3}\sqrt{2}. \] 3. Notice that the terms \(\sqrt{2}\sqrt{3}\) cancel: \[ \sqrt{2}\sqrt{3} - \sqrt{3}\sqrt{2} = 0. \] 4. The remaining terms are: \[ -(\sqrt{2})^2 + (\sqrt{3})^2 = -2 + 3 = 1. \] 5. Multiply by the factor \(2\): \[ 2 \cdot 1 = 2. \] Thus, \[ (\sqrt{2}+\sqrt{3})(2\sqrt{3}-2\sqrt{2}) = 2. \] --- **2.1.2** Simplify \[ \sqrt[y]{\frac{6^y+5\cdot 3^y}{10^y+5^{y+1}}}. \] 1. Rewrite the numerator: \[ 6^y + 5\cdot 3^y = (2\cdot 3)^y + 5\cdot 3^y = 2^y\cdot 3^y + 5\cdot 3^y. \] Factor out \(3^y\): \[ 3^y (2^y + 5). \] 2. Rewrite the denominator: \[ 10^y + 5^{y+1} = (2\cdot 5)^y + 5\cdot 5^y = 2^y\cdot 5^y + 5\cdot 5^y. \] Factor out \(5^y\): \[ 5^y (2^y + 5). \] 3. Form the fraction: \[ \frac{3^y (2^y+5)}{5^y (2^y+5)}. \] Cancel the common factor \((2^y+5)\) (assuming \(2^y+5 \neq 0\)): \[ \frac{3^y}{5^y} = \left(\frac{3}{5}\right)^y. \] 4. Now, take the \(y\)th root: \[ \sqrt[y]{\left(\frac{3}{5}\right)^y} = \left(\left(\frac{3}{5}\right)^y\right)^{\frac{1}{y}} = \frac{3}{5}. \] Thus, \[ \sqrt[y]{\frac{6^y+5\cdot 3^y}{10^y+5^{y+1}}} = \frac{3}{5}. \]