Problem 103 - 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. $$ \frac{480,000,000,000}{0.00012} $$

Step-by-Step Solution

Answer

In scientific notation, the result of the division operation is \(4 \times 10^{15}\).

Step 1: Perform the Division

First, divide 480,000,000,000 by 0.00012. This results in \(4 \times 10^{15}\).

Step 2: Write the Result in Scientific Notation

The number 4,000,000,000,000,000 can be written in scientific notation as \(4 \times 10^{15}\). Here, 4 is the decimal factor and 15 is the exponent.

Step 3: Round the Decimal Factor

As the decimal factor in the scientific notation answer is 4, which is already a number with two decimal places, there is no need for a rounding step.

Key Concepts

Division

Division is a mathematical operation where a larger number, called the dividend, is divided by a smaller number, called the divisor. It simplifies the expression, and in this exercise, helps us understand how a large number (\(480,000,000,000\)) can be divided by a very small one (\(0.00012\)).
When you divide - A large number by a small number, you typically get an even larger number.- For example, dividing \(480,000,000,000\) by \(0.00012\) initially seems complex, but it can be simplified using scientific notation.In practice, it's often useful to convert both numbers into scientific notation first, performing the division step by step to simplify.
Understanding division is crucial since it forms the basis for many operations, and recognizing how division affects the magnitude is particularly handy, especially in scientific and real-life applications.

Exponents

Exponents represent repeated multiplication of a number by itself. For instance, \(10^{15}\) means the number 10 multiplied by itself 15 times. This compact form, known as an exponential notation, makes it easier to express very large or small numbers.
In scientific notation, exponents help convey the scale efficiently.

• The exponent shows the number of places to move the decimal point.
• Positive exponents move the decimal to the right; negative ones do the opposite.

In this exercise, the division resulted in \(4 \times 10^{15}\), meaning that the original number was so large that you must move 15 places to reach the correct digit.
Understanding exponents simplifies working with vast datasets and is the backbone of representing numbers in scientific notation.

Rounding Decimals

Rounding decimals makes numbers easier to handle by trimming extra digits while maintaining a desired level of precision.
In scientific notation, the decimal factor (the number multiplied by the power of ten) might need rounding to simplify results.

• If you round to two decimal places, observe the third decimal.
• If the third decimal is 5 or more, increase the second decimal by one; otherwise, leave it.

In this example, the decimal factor was \(4\), so no rounding was necessary. But generally, be conscious of accuracy when rounding, as it can impact calculations, especially in science and engineering.

Decimal Factor

The decimal factor is the number in scientific notation that directly precedes the multiplication by a power of ten.
This factor reflects our main quantity while the exponent indicates its scale. Hence, \(4\) (in \(4 \times 10^{15}\)) is the decimal factor, acting as the main number being modified by the power of ten.

• Typically a number between 1 and 10.
• Expresses the significant figures of the original number.

The decimal factor's role holds the essence of the original measurement or calculation, providing a clearer picture without overwhelming detail.Understanding its importance helps in simplifying any further mathematical processes and report accuracy in complex equations or large datasets.