2.3 Solve the following equations. 2.3.1 \( \quad 3^{x}=81 \) 2.3.2 \( \quad 5^{x}=\frac{1}{25} \) \( 2.3 .3 \quad 2^{x+4}=1 \) 2.4 Calculate the following and give your final answer in scientific notation. \[ 2,5 \times 10^{-6} \times 6 \times 10^{10}+5,7 \times 10^{8} \] [21] Question 3 3.1 You receive R2500 from your grandfather on your birthday. You decide to invest the money in the bank for you studies four years from now. The bank gives you the following two options. OPTION A: Invested at 8\% compound interest pa for the four years. OPTIONB: Invested at 7,5\% simple interest pa for the four years. Decide which one will generate more money in the give period. 3.2 Calculate the percentage increase if the price of diesel increases from R11,21 per litre to R12,30 per litre. Round your answer off to 2 decimal places. 3.3 An aircraft travels at a speed of \( 0,9 \times 10^{3} \mathrm{~km} / \mathrm{h} \) for 24 hours. How far has the aircraft travelled? Give the answer in scientific notation. (7) [14]

To solve \(3^{x}=81\), we recognize that \(81\) can be expressed as \(3^{4}\). Hence, we can set up the equation \(3^{x}=3^{4}\), leading to \(x=4\). For \(5^{x}=\frac{1}{25}\), we realize that \(\frac{1}{25}\) equals \(5^{-2}\). This gives us \(5^{x}=5^{-2}\), so \(x=-2\). When considering \(2^{x+4}=1\), we recognize that \(1\) can be expressed as \(2^{0}\), leading to \(x+4=0\) and therefore \(x=-4\). Now, onto the calculations! To compute \(2.5 \times 10^{-6} \times 6 \times 10^{10}\), we multiply \(2.5 \times 6 = 15\) and combine the powers of ten, giving \(15 \times 10^{4}\). Adding \(5.7 \times 10^{8}\) requires expressing \(15 \times 10^{4}\) in the same order of magnitude. It becomes \(0.00015 \times 10^{8}\). Thus, adding gives \(5.70015 \times 10^{8}\). In scientific notation, this is approximately \(5.70 \times 10^{8}\). Now, let's dig into the investment options! Option A, with compound interest, can be calculated using the formula \(A = P(1 + r)^t\); substituting the values gives us \(A = 2500(1 + 0.08)^{4} = 2500(1.36049) \approx R3401.22\). In contrast, Option B employs simple interest calculated via \(A = P(1 + rt)\), yielding \(A = 2500(1 + 0.075 \cdot 4) = 2500(1.3) = R3250\). Clearly, Option A beats Option B! As for diesel price increase calculations, we can find the percentage increase using the formula \(\text{Percentage Increase} = \frac{\text{New Price} - \text{Old Price}}{\text{Old Price}} \times 100\). Substituting the values gives \(\frac{12.30 - 11.21}{11.21} \times 100 \approx 9.73\%\). Lastly, for the aircraft's distance calculation, at \(0.9 \times 10^3 \text{ km/h}\) flying for 24 hours means the distance is \((0.9 \times 10^3) \times 24 = 21.6 \times 10^3 \text{ km}\), or \(2.16 \times 10^{4} \text{ km}\) in scientific notation.