\( \begin{array}{l}\text { 1. Given a number written in scientific notation, how } \\ \text { do you adjust the exponent and decimal if needed? } \\ \text { Evaluate the following. } \\ \begin{array}{ll}\left(6.3 \times 10^{2}\right)+\left(4.1 \times 10^{3}\right) & \text { 3. }\left(9.2 \times 10^{-5}\right)-\left(2.4 \times 10^{-6}\right) \\ \left(10^{2} \times 10^{3}\right)+(4.1 \times 6.3) \\ 10^{5}+25.83 & \text { 5. }\left(1.9 \times 10^{-2}\right) \div\left(2 \times 10^{-9}\right)\end{array} \\ \begin{array}{l}\text { 4. }\left(7.2 \times 10^{-7}\right) \times\left(5.8 \times 10^{4}\right)\end{array}\end{array} \)

To solve the problems involving numbers in scientific notation, we will follow these steps: 1. **Adjust the exponents and decimals if needed**: When adding or subtracting numbers in scientific notation, the exponents must be the same. If they are not, we adjust one of the numbers to match the exponent of the other. For multiplication and division, we can directly multiply or divide the coefficients and add or subtract the exponents. 2. **Evaluate each expression step by step**. Let's evaluate each expression one by one. ### 1. \( (6.3 \times 10^{2}) + (4.1 \times 10^{3}) \) To add these two numbers, we need to adjust the exponents: - Convert \( 4.1 \times 10^{3} \) to the same exponent as \( 6.3 \times 10^{2} \): \[ 4.1 \times 10^{3} = 4.1 \times 10^{1} \times 10^{2} = 41.0 \times 10^{2} \] Now we can add: \[ (6.3 + 41.0) \times 10^{2} = 47.3 \times 10^{2} \] ### 2. \( (9.2 \times 10^{-5}) - (2.4 \times 10^{-6}) \) Here, we need to adjust the exponents again: - Convert \( 2.4 \times 10^{-6} \) to the same exponent as \( 9.2 \times 10^{-5} \): \[ 2.4 \times 10^{-6} = 0.24 \times 10^{-5} \] Now we can subtract: \[ (9.2 - 0.24) \times 10^{-5} = 8.96 \times 10^{-5} \] ### 3. \( (10^{2} \times 10^{3}) + (4.1 \times 6.3) \) For the first part: \[ 10^{2} \times 10^{3} = 10^{2 + 3} = 10^{5} \] For the second part: \[ 4.1 \times 6.3 = 25.83 \] Now we can add: \[ 10^{5} + 25.83 \] Since \( 10^{5} \) is much larger than \( 25.83 \), we can express the result as: \[ 10^{5} + 25.83 \approx 10^{5} \text{ (dominant term)} \] ### 4. \( (7.2 \times 10^{-7}) \times (5.8 \times 10^{4}) \) For multiplication: \[ (7.2 \times 5.8) \times (10^{-7} \times 10^{4}) = 41.76 \times 10^{-3} \] ### 5. \( (1.9 \times 10^{-2}) \div (2 \times 10^{-9}) \) For division: \[ \frac{1.9}{2} \times 10^{-2 - (-9)} = 0.95 \times 10^{7} \] Now, let's summarize the results: 1. \( (6.3 \times 10^{2}) + (4.1 \times 10^{3}) = 47.3 \times 10^{2} \) 2. \( (9.2 \times 10^{-5}) - (2.4 \times 10^{-6}) = 8.96 \times 10^{-5} \) 3. \( (10^{2} \times 10^{3}) + (4.1 \times 6.3) = 10^{5} + 25.83 \) 4. \( (7.2 \times 10^{-7}) \times (5.8 \times 10^{4}) = 41.76 \times 10^{-3} \) 5. \( (1.9 \times 10^{-2}) \div (2 \times 10^{-9}) = 0.95 \times 10^{7} \) These are the evaluated results for each expression.