QUESTION 2 2.1 Simplify fully without the use of a calculator: \[ \frac{2^{m+5}-3.2^{m-2}}{5.2^{m+3}} \] 2.2 Solve for \( x \) : \[ 4^{x}-5.2^{x}-24=0 \] 2.3 If \( x=\frac{3-\sqrt{a}}{\sqrt{2}} \) and \( y=\frac{4+\sqrt{a}}{\sqrt{2}} \), determine the value of \( (x+y)^{2} \).

To simplify the expression, consider the structure of the terms involved. In the numerator, you can factor out \( 2^{m-2} \) to make the calculation clearer: \[ \frac{2^{m+5}-3 \cdot 2^{m-2}}{5 \cdot 2^{m+3}} = \frac{2^{m-2}(2^{7}-3)}{5 \cdot 2^{m+3}} = \frac{2^{m-2}(128-3)}{5 \cdot 2^{m+3}} = \frac{125 \cdot 2^{m-2}}{5 \cdot 2^{m+3}} = \frac{125}{5 \cdot 2^{5}} = \frac{25}{32} \] But don't stop there! When you simplify further, you find that it's always useful to remember that fractions are just as tasty as cakes: stack them wisely! Now, for the second part, rearranging our equation helps. Replacing \(4^x\) with \( (2^2)^x = 2^{2x} \) allows for a neat transformation: \[ 2^{2x} - 5 \cdot 2^{x} - 24 = 0 \] Then, substituting \(y = 2^{x}\) turns the equation into a standard quadratic: \[ y^2 - 5y - 24 = 0 \] Using the quadratic formula gives us \(y = 8\) or \(y = -3\). Since \(y = 2^x\) must be a positive value, we pick \(y=8\), so \(x=3\). Fun fact: Quadratic equations often remind us of those delightful mystery puzzles! The solutions are like clues that guide you to the treasure! Lastly, for \(x+y\): \[ x+y = \frac{3-\sqrt{a}}{\sqrt{2}} + \frac{4+\sqrt{a}}{\sqrt{2}} = \frac{(3-\sqrt{a}) + (4+\sqrt{a})}{\sqrt{2}} = \frac{7}{\sqrt{2}} \] Calculating \((x+y)^2\): \[ (x+y)^2 = \left(\frac{7}{\sqrt{2}}\right)^2 = \frac{49}{2} \] When you're working with squares, remember: they’re not just a shape; they bring symmetry to the mathematical party!