Problem 33 - Elementary Algebra

Question

Convert the numbers used in the following problems to scientific notation. The mass of the Eiffel tower in Paris, France, is 8,000,000 grams.

Step-by-Step Solution

Answer

Answer: The mass of the Eiffel Tower in scientific notation is 8.0 x 10^6 grams.

Step 1: Find the value of A

To find the value of A, we need to move the decimal point in the given number, 8,000,000 grams, to the left until it is between the first and the second non-zero digits. In this case, move the decimal until it is between the 8 and the first 0, which gives us 8.0. So, the value of A is 8.0.

Step 2: Count the number of times the decimal point is moved

Now, count how many times we had to move the decimal point in Step 1. We moved it 6 times to the left. Therefore, the value of B is 6.

Step 3: Write the number in scientific notation

In scientific notation, the number should be expressed as A x 10^B. We have found A to be 8.0 and B to be 6. Therefore, we can write the mass of the Eiffel Tower in scientific notation as: 8.0 x 10^6 grams

Key Concepts

Converting to Scientific Notation

Understanding how to convert large or small numbers into scientific notation simplifies complex calculations and helps to standardize measurements in science and engineering. Scientific notation expresses numbers as a product of two parts: a coefficient and a power of 10. The coefficient, often referred to as the base, must be a number greater than or equal to 1 and less than 10. The power of 10 is written as an exponent.

To convert a number to scientific notation, first identify the coefficient by placing the decimal point after the first non-zero digit. As evidenced in the exercise above, the number 8,000,000 grams became 8.0 once the decimal was moved. Next, determine the exponent by counting the number of spaces the decimal point was shifted. In the provided example, it was moved 6 spaces, resulting in an exponent of 6. Finally, combine the coefficient and the exponent to finish the conversion: 8.0 x 10^6 grams.

Significant Figures

Significant figures, or 'sig figs', are critical in scientific measurements as they indicate the precision of a measurement or calculation. They include all known digits plus one estimated digit, giving an insight into the reliability of a number. To identify significant figures, start from the first non-zero digit, counting that and every digit to its right as significant.

In the exercise's number 8,000,000 grams, the coefficient after conversion is 8.0, which contains two significant figures: the 8 and the zero following the decimal point. The zero indicates the precision of the measurement up to the hundredths place. When converting to scientific notation, it’s important to maintain the same number of significant figures as the original number to ensure that precision is neither lost nor artificially enhanced.

Exponent Form

The exponent form, often associated with scientific notation, is pivotal for representing very large or small numbers succinctly. An exponent indicates how many times a base number is multiplied by itself. For instance, in the scientific notation of 8.0 x 10^6 grams, the exponent is 6, which means the base number (10) is to be multiplied by itself 6 times (10 x 10 x 10 x 10 x 10 x 10).

The exponent serves as a 'compact' shorthand that removes the need for writing out extensive strings of zeros in long numbers. This is especially useful when dealing with quantities in astronomy, physics, and other scientific fields where extreme ranges of values are common. Remember, a positive exponent indicates a large number, while a negative exponent points to a small one, symbolizing division (for example, 10^-6 means dividing 1 by 10 six times).