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. OPTION B: Invested at 7,5\% simple interest pa for the four years. Decide which one will generate more money in the give period. (7) 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. (4) travelled? Give the answer in scientific notation. (3) [14]

Calculate the value by following steps: - step0: Calculate: \(0.9\times 10^{3}\times 24\) - step1: Multiply the terms: \(21.6\times 10^{3}\) - step2: Rewrite the expression: \(21.6\times 1000\) - step3: Multiply the numbers: \(21600\) Calculate or simplify the expression \( 2500 \times (1 + 0.075 \times 4) \). Calculate the value by following steps: - step0: Calculate: \(2500\left(1+0.075\times 4\right)\) - step1: Multiply the numbers: \(2500\left(1+0.3\right)\) - step2: Add the numbers: \(2500\times 1.3\) - step3: Multiply the numbers: \(3250\) Calculate or simplify the expression \( 2.5 \times 10^{-6} \times 6 \times 10^{10} + 5.7 \times 10^{8} \). Calculate the value by following steps: - step0: Calculate: \(2.5\times 10^{-6}\times 6\times 10^{10}+5.7\times 10^{8}\) - step1: Multiply the terms: \(15\times 10^{4}+5.7\times 10^{8}\) - step2: Evaluate the power: \(150000+5.7\times 10^{8}\) - step3: Rewrite the expression: \(150000+570000000\) - step4: Add the numbers: \(570150000\) Solve the equation \( 5^{x}=\frac{1}{25} \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{x}=\frac{1}{25}\) - step1: Rewrite in exponential form: \(5^{x}=5^{-2}\) - step2: Set the exponents equal: \(x=-2\) Calculate or simplify the expression \( (12.30 - 11.21) / 11.21 * 100 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(12.3-11.21\right)}{11.21}\times 100\) - step1: Remove the parentheses: \(\frac{12.3-11.21}{11.21}\times 100\) - step2: Subtract the numbers: \(\frac{1.09}{11.21}\times 100\) - step3: Divide the terms: \(\frac{109}{1121}\times 100\) - step4: Multiply: \(\frac{109\times 100}{1121}\) - step5: Multiply: \(\frac{10900}{1121}\) Solve the equation \( 3^{x}=81 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x}=81\) - step1: Rewrite in exponential form: \(3^{x}=3^{4}\) - step2: Set the exponents equal: \(x=4\) Solve the equation \( 2^{x+4}=1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{x+4}=1\) - step1: Rewrite in exponential form: \(2^{x+4}=2^{0}\) - step2: Set the exponents equal: \(x+4=0\) - step3: Move the constant to the right side: \(x=0-4\) - step4: Remove 0: \(x=-4\) Calculate or simplify the expression \( 2500 \times (1 + 0.08)^{4} \). Calculate the value by following steps: - step0: Calculate: \(2500\left(1+0.08\right)^{4}\) - step1: Add the numbers: \(2500\times 1.08^{4}\) - step2: Convert the expressions: \(2500\left(\frac{27}{25}\right)^{4}\) - step3: Simplify: \(2500\times \frac{27^{4}}{25^{4}}\) - step4: Rewrite the expression: \(625\times 4\times \frac{27^{4}}{25^{4}}\) - step5: Rewrite the expression: \(25^{2}\times 4\times \frac{27^{4}}{25^{4}}\) - step6: Reduce the numbers: \(4\times \frac{27^{4}}{25^{2}}\) - step7: Multiply: \(\frac{4\times 27^{4}}{25^{2}}\) - step8: Simplify: \(\frac{4\times 27^{4}}{625}\) Let's solve the problems step by step. ### 2.3 Solve the following equations. #### 2.3.1 \( 3^{x} = 81 \) To solve for \( x \), we can express 81 as a power of 3: \[ 81 = 3^4 \] Thus, we have: \[ 3^{x} = 3^{4} \] This implies: \[ x = 4 \] #### 2.3.2 \( 5^{x} = \frac{1}{25} \) We can express \( \frac{1}{25} \) as a power of 5: \[ \frac{1}{25} = 5^{-2} \] Thus, we have: \[ 5^{x} = 5^{-2} \] This implies: \[ x = -2 \] #### 2.3.3 \( 2^{x+4} = 1 \) Since \( 1 \) can be expressed as \( 2^0 \): \[ 2^{x+4} = 2^{0} \] This implies: \[ x + 4 = 0 \] Thus: \[ x = -4 \] ### 2.4 Calculate the following and give your final answer in scientific notation. \[ 2.5 \times 10^{-6} \times 6 \times 10^{10} + 5.7 \times 10^{8} \] Calculating this gives: \[ 2.5 \times 6 = 15 \] So, \[ 15 \times 10^{4} + 5.7 \times 10^{8} \] Converting \( 15 \times 10^{4} \) to scientific notation: \[ 1.5 \times 10^{5} \] Now we can add: \[ 1.5 \times 10^{5} + 5.7 \times 10^{8} = 5.7015 \times 10^{8} \] ### Question 3 #### 3.1 Investment Options **OPTION A:** \[ A = P(1 + r)^n = 2500(1 + 0.08)^4 \] Calculating this gives: \[ A \approx 3401.22 \] **OPTION B:** \[ A = P(1 + rt) = 2500(1 + 0.075 \times 4) \] Calculating this gives: \[ A = 2500 \times 1.3 = 3250 \] **Conclusion:** OPTION A generates more money, approximately R3401.22 compared to R3250. #### 3.2 Percentage Increase The price of diesel increases from R11.21 to R12.30. The percentage increase is calculated as: \[ \text{Percentage Increase} = \frac{(12.30 - 11.21)}{11.21} \times 100 \approx 9.72\% \] #### 3.3 Distance Travelled by Aircraft The aircraft travels at a speed of \( 0.9 \times 10^{3} \) km/h for 24 hours: \[ \text{Distance} = \text{Speed} \times \text{Time} = 0.9 \times 10^{3} \times 24 = 21600 \text{ km} \] In scientific notation: \[ 2.16 \times 10^{4} \text{ km} \] ### Summary of Answers - 2.3.1: \( x = 4 \) - 2.3.2: \( x = -2 \) - 2.3.3: \( x = -4 \) - 2.4: \( 5.7015 \times 10^{8} \) - 3.1: OPTION A generates more money (R3401.22) - 3.2: Percentage increase is approximately \( 9.72\% \) - 3.3: Distance travelled is \( 2.16 \times 10^{4} \) km