Problem 58 - Algebra 1: Concepts and Skills

Question

Use a calculator to perform the indicated operation. Write the result in scientific notation and in decimal form. $$ \frac{3,940,000}{0.0002} $$

Step-by-Step Solution

Answer

The result of the division operation is \(19,700,000,000\) in decimal form and in scientific notation, it is \(1.97 \times 10^{10}\).

Step 1: Perform the Division

Divide 3,940,000 by 0.0002 using a calculator. The result obtained is \(19,700,000,000\) (nineteen billion seven hundred million).

Step 2: Write the Result in Standard Form

The result, nineteen billion seven hundred million, in standard form is \(19,700,000,000\).

Step 3: Convert to Scientific Notation

To convert the result into scientific notation, move the decimal point to the right of the first non-zero digit. There are 10 digits to the left of the decimal, ​so the power of 10 becomes 10 in the scientific notation. Thus, in scientific notation, the result is expressed as \(1.97 \times 10^{10}\).

Key Concepts

Performing Divisions with a Calculator

When you're tasked with solving a mathematical problem involving division, especially with big numbers or numbers with many decimal places, using a calculator can save you a lot of time and ensure accuracy. The process of performing a division on a calculator is straightforward.

First, you should key in the numerator (the number you're dividing) followed by the division symbol. Next, you enter the denominator (the number you're dividing by) and press the equal sign. Always double-check that you've entered the numbers correctly to avoid mistakes. For our example, dividing 3,940,000 by 0.0002 gives a large number, which the calculator displays as 19,700,000,000.

Writing Numbers in Decimal Form

Numbers can be expressed in various forms, and decimal form is one of the most conventional and widely used methods. Writing a number in decimal form means expressing it as a sequence of digits with a decimal point to separate the whole number part from the fractional part, if any.

In cases where the number is a whole number like our resulting 19,700,000,000 from the division operation, there's no fractional part, and hence no decimal point is visible. However, it's understood that the decimal point follows immediately after the last zero. Make sure to write out the entire number, even if it's large, to accurately reflect its value in decimal form.

Converting to Scientific Notation

Scientific notation is a method of writing numbers that accommodates values that are too large or too small to be conveniently written in decimal form. This notation expresses numbers as a product of two factors: a coefficient that is at least 1 but less than 10, and a power of 10.

To convert a large number like 19,700,000,000 to scientific notation, identify the first non-zero digit and place the decimal point there. Count how many places you move the decimal point until it is after the first digit – in this case, it's 10 places. The number 19,700,000,000 becomes 1.97 when trimmed down like this. Since we moved the decimal point 10 places, we multiply 1.97 by 10 to the power of 10, writing it as \(1.97 \times 10^{10}\) in scientific notation. Writing numbers in scientific notation makes them easier to work with, especially in calculations involving very large or small numbers.