Problem 95 - Algebra and Trigonometry

Question

Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$ \frac{8.4 \times 10^{8}}{4 \times 10^{5}} $$

Step-by-Step Solution

Answer

The solution to the exercice is \(2.1 \times 10^{3}\).

Step 1: Dividing the coefficients

The first step is to divide the coefficients (numbers before the power of 10). In this case, the coefficients are 8.4 and 4. So, perform the division: \( \frac{8.4}{4} = 2.1 \)

Step 2: Subtracting the exponents

Next, we must subtract the exponents of 10 from each other. Here, the exponents are \(10^{8}\) and \(10^{5}\). So, we subtract \(8-5 = 3\) which means \(10^{8} ÷ 10^{5} = 10^{3} \)

Step 3: Combine the results

Lastly, we combine the results from step 1 and step 2 to give the final answer in scientific notation. The coefficient is 2.1 and the power of 10 is 3. Therefore, the result is \(2.1 \times 10^{3} \)

Key Concepts

Coefficients

In scientific notation, the coefficient is an important part of the expression. It is the number that comes before the multiplication sign and the power of 10. Whenever you are performing calculations in scientific notation, the first step is to handle these coefficients. For example, in the expression \(\frac{8.4 \times 10^{8}}{4 \times 10^{5}}\), the coefficients are 8.4 and 4. To simplify the operation, you should divide the coefficients. The process involves simple division:

• Divide the numbers as you would in regular arithmetic.
• Using a calculator or long division, \(\frac{8.4}{4}\) equals 2.1.

This result will be the new coefficient in the final scientific notation. It’s crucial as it determines the scale of the number, ensuring the expression correctly represents the original value.

Exponents

Exponents in scientific notation indicate how many times a number should be multiplied by 10. Managing exponents is essential when multiplying or dividing numbers in scientific notation. In the expression \(\frac{8.4 \times 10^{8}}{4 \times 10^{5}}\), the exponents are 8 and 5. To simplify this, we handle the division of exponential terms by subtracting the exponent of the divisor from that of the dividend. Here are the key steps to follow:

• Identify the exponents attached to the power of 10 in both terms. In this case, they are 8 and 5.
• Subtract the exponent of the denominator from the exponent of the numerator: \(8 - 5 = 3\).

This gives us a new exponent, 3, and results in the power of 10 for the simplified expression being \(10^{3}\). Managing exponents correctly helps maintain accuracy and proportionality in the representation.

Rounding Decimals

Rounding decimals is often necessary when dealing with scientific notation, especially when precision is crucial or specified. In scientific notation problems, you might be asked to round the coefficient to a certain number of decimal places. If needed, rounding improves readability and conforms the result to specific guidelines. Consider the expression \(2.1 \times 10^{3}\):

• For rounding, determine the number of decimal places to round to, such as two decimal places.
• Inspect the digit immediately after your desired decimal place. If it's 5 or higher, round up. If it's lower, keep the current value.

In our final answer, the coefficient is already at one decimal place. However, if it were longer, you would apply these principles to adjust it to two decimal places. This ensures the result is not just mathematically correct, but also easy to communicate and understand.