Problem 93 - 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. $$ \left(4.3 \times 10^{8}\right)\left(6.2 \times 10^{4}\right) $$

Step-by-Step Solution

Answer

The final answer is \(2.67 \times 10^{13}\)

Step 1: Multiply Significant Figures

Begin by multiplying the significant figures together. In this case, multiply 4.3 by 6.2 to get 26.66.

Step 2: Add Exponents

Next, add together the powers of ten from each number. In this case, add 8 and 4 together to get 12.

Step 3: Combine Results in Scientific Notation

Combine the results from Step 1 and Step 2 to express the answer in scientific notation. The answer is \(26.66 \times 10^{12}\).

Step 4: Express Result Correctly

The number 26.66 is not between 1 and 10, so it's necessary to adjust the significant figure and exponent. By moving the decimal place one place to the left, we get 2.666 which rounds to 2.67 (to keep two decimal places) and we have to add one to the exponent (since we made the significant figure 10 times smaller), making it 13. Hence, the final answer is \(2.67 \times 10^{13}\).

Key Concepts

Multiplying Significant Figures

When multiplying numbers that are in scientific notation, it is crucial to first address the significant figures. Significant figures are the digits in a number that contribute to its precision. In the given problem, we have the numbers 4.3 and 6.2. These numbers have two significant figures each. To begin, multiply these significant figures together:

• 4.3 times 6.2 equals 26.66.

The result 26.66 initially appears to have four significant figures. However, when dealing with multiplying significant figures, the final product should contain the same number of significant figures as the number with the fewest significant figures in the problem. Since both numbers have two, we will consider them as such in further steps.

Adding Exponents

In scientific notation, numbers are expressed as a product of a number (the significant figure) and a power of ten. For the provided numbers, 10 is raised to an exponent, known as the power of ten. In this example, we have:

• 10 raised to the power of 8
• 10 raised to the power of 4

To multiply these powers of ten, simply add the exponents together:

• 8 plus 4 equals 12.

Thus, the cumulative effect of the powers of ten is represented as 10 raised to the power of 12. Combining this with our earlier multiplication gives us a raw result of 26.66 times 10 raised to 12.

Rounding Decimals

The result from the multiplication and addition steps is 26.66 times 10 raised to the power of 12. However, scientific notation requires the coefficient (that is, the significant figure) to be between 1 and 10. Since 26.66 is not, we must adjust it. By moving the decimal point one place to the left, 26.66 becomes 2.666. This operation decreases its magnitude, so we compensate by increasing the exponent by one, getting 10 raised to the power of 13. Next, we round 2.666 to the appropriate number of significant figures, which is two in this case. Therefore, we round it to 2.67. This new rounded number maintains the integrity of the original precision while fitting scientific notation conventions.

Expressing in Scientific Notation

Finally, we consolidate our results to present the final answer in proper scientific notation form. The goal is to have a number between 1 and 10 multiplied by a power of ten. Having performed the necessary multiplication, exponent addition, and rounding steps, we now express the result coherently:- The adjusted significant figure is 2.67.- The adjusted power of ten is 13.Thus, the number is written in scientific notation as:\[2.67 \times 10^{13}\]This format clearly and concisely conveys the number, maintaining all necessary precision and adhering to scientific notation standards.