2.1 Simplify fully, WITHOUT using a calculator \( \frac{3^{m+1}-6.3^{m+1}}{7.3^{m+2}} \) Solve for \( x \), WITHOUT using a calculator: \( 2.2 .1 \quad x^{-\frac{2}{4}}=8 \) \( 2.2 .2 \quad 4^{x}-2^{x}=2 \) If \( x=\frac{3-\sqrt{a}}{\sqrt{2}} \) and \( y=\frac{4+\sqrt{a}}{\sqrt{2}} \), determin 4 Show, WITHOUT using a calculator, that

To simplify the expression \( \frac{3^{m+1}-6.3^{m+1}}{7.3^{m+2}} \), start by factoring out the numerator. Recognizing that \( 6 \) can be written as \( 2 \times 3 \), rewrite the numerator as \( 3^{m+1}(1 - 2) = 3^{m+1}(-1) = -3^{m+1} \). Then simplify the denominator \( 7.3^{m+2} \) to \( 7.3^{m+1} \times 3 \). Finally, your expression simplifies to \( \frac{-3^{m+1}}{7.3^{m+1} \times 3} = -\frac{1}{7} \). To solve \( x^{-\frac{2}{4}}=8 \), first re-write the exponent: \( x^{-0.5} = 8 \) which can be flipped to become \( \frac{1}{\sqrt{x}} = 8 \). Next, multiply both sides by \( \sqrt{x} \) and get \( 1 = 8\sqrt{x} \). Dividing by 8 results in \( \sqrt{x} = \frac{1}{8} \), then squaring both sides gives \( x = \frac{1}{64} \). Now for the equation \( 4^{x} - 2^{x} = 2 \). Since \( 4^{x} = (2^2)^{x} = 2^{2x} \), you can rewrite the equation as \( 2^{2x} - 2^x - 2 = 0 \). Let \( u = 2^x \), transforming it into a quadratic: \( u^2 - u - 2 = 0 \). Factoring gives \( (u-2)(u+1) = 0\), yielding solutions \( u = 2 \) or \( u = -1 \) (discarding the negative root). Hence, \( 2^x = 2 \) leads to \( x = 1 \). Lastly, for the expressions \( x=\frac{3-\sqrt{a}}{\sqrt{2}} \) and \( y=\frac{4+\sqrt{a}}{\sqrt{2}} \), if we multiply them together, we get \( xy = \frac{(3-\sqrt{a})(4+\sqrt{a})}{2} = \frac{12 - 3\sqrt{a} + 4\sqrt{a} - a}{2} = \frac{12 + \sqrt{a} - a}{2}\). This shows how to connect values and establish relationships between \(x\) and \(y\) creatively!