Problem 32 - Elementary Algebra

Question

Convert the numbers used in the following problems to scientific notation. The largest brain ever measured was that of a sperm whale. It had a mass of 9200 grams.

Step-by-Step Solution

Answer

Question: Convert the mass of a sperm whale's brain (9200 grams) to scientific notation. Answer: \(9.2 \times 10^2\) grams.

Step 1: Identify the number to be converted

The number given is the mass of the sperm whale's brain, which is 9200 grams.

Step 2: Convert the number to scientific notation

To convert the number 9200 to scientific notation, first find a number between 1 and 10 by moving the decimal point two places to the left. This gives 9.2. Now, since the decimal point was moved two places to the left, multiply 9.2 by 10 raised to the power of 2 (which is the number of places moved). So, the scientific notation for 9200 is: \(9.2 \times 10^2\) The mass of the sperm whale's brain in scientific notation is \(9.2 \times 10^2\) grams.

Key Concepts

Exponents

Exponents are an essential concept in mathematics, especially when dealing with scientific notation. Simply put, an exponent refers to the number of times a number, known as the base, is multiplied by itself. In the context of scientific notation, exponents are used particularly with the base of 10. This is because scientific notation expresses numbers as a product of a coefficient and a power of 10.

For instance, in scientific notation, the number 9200 becomes 9.2 multiplied by 10 raised to the power of 2, which can be written as \(9.2 \times 10^2\). The exponent 2 tells us that 10 is multiplied by itself once, making it 100. Therefore, when 9.2 is multiplied by 100, it results in 920, which is the number of times we remember moving the decimal point in the process.

Decimal Point

Understanding how to move the decimal point is crucial when converting numbers into scientific notation. The decimal point is the small dot in a number that separates the whole number from the fractions. In whole numbers like 9200, the decimal point is at the end, although it's often invisible.

• In 9200, consider the number as 9200.0, where the decimal point is after the zero.
• Moving the decimal point two places to the left, you get 9.2.

Such shifting of the decimal is important because it helps in expressing large (or small) numbers more compactly. In this manner, significant figures are easily represented, and the zeros are accounted for by using an appropriate power of 10.

Conversion to Scientific Notation

Conversion to scientific notation involves two main steps: finding a number between 1 and 10 and identifying the exponent. Let's break this down further.

• First, you need to create a number between 1 and 10 from the original number. For 9200, moving the decimal two places to the left makes 9.2.
• Second, determine how many places the decimal has moved. This number becomes your exponent. In this case, because we moved two places, the exponent is 2.

Finally, you combine these findings: the coefficient (9.2) and the power of 10 (\(10^2\)). Therefore, 9200 expressed in scientific notation is \(9.2 \times 10^2\). This method of conversion makes it simpler to handle extremely large or small numbers in a concise form.