2.1 Simplify WITHOUT using a calculator: 2.1.1 \( \left(2^{-1}+3^{-1}\right)^{2} \) 2.1.2 \( \frac{2.4^{x+2}-4^{x-3.7}}{2^{x} \cdot 2^{x}} \) 2.1.3 \( 3^{-\frac{1}{2}}[\sqrt{12}+\sqrt[3]{3 \sqrt{3}}] \)
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To simplify the expressions, let's break it down step by step: **2.1.1** Start with the expression inside the parentheses: \( 2^{-1} + 3^{-1} = \frac{1}{2} + \frac{1}{3} \). Finding a common denominator (which is 6), we have \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \). Now, square this result: \[ \left( \frac{5}{6} \right)^{2} = \frac{25}{36}. \] **2.1.2** First, rewrite \( 4^{x-3.7} \) as \( (2^{2})^{x-3.7} = 2^{2x - 7.4} \). The original expression becomes: \[ \frac{2.4^{x+2} - 2^{2x - 7.4}}{2^{x} \cdot 2^{x}} = \frac{2^{x+2} - 2^{2x - 7.4}}{2^{2x}}. \] This simplifies to \( \frac{2^{x+2}}{2^{2x}} - \frac{2^{2x - 7.4}}{2^{2x}} = 2^{2-x} - 2^{-7.4} = 2^{2-x} - \frac{1}{2^{7.4}} \). **2.1.3** Start with the term \( 3^{-\frac{1}{2}} = \frac{1}{\sqrt{3}} \). Now calculate \( \sqrt{12} = 2\sqrt{3} \) and \( \sqrt[3]{3\sqrt{3}} = 3^{\frac{2}{3}} \). Expressing these together gives: \[ \sqrt{12} + \sqrt[3]{3 \sqrt{3}} = 2\sqrt{3} + 3^{\frac{2}{3}}. \] Finally, multiply by \( \frac{1}{\sqrt{3}} \): \[ 3^{-\frac{1}{2}}(2\sqrt{3} + 3^{\frac{2}{3}}) = \frac{2\sqrt{3}}{\sqrt{3}} + \frac{3^{\frac{2}{3}}}{\sqrt{3}} = 2 + 3^{\frac{1}{3}}. \]
