Problem 13 - Algebra 1

Question

Rewrite in scientific notation. $$ 72,000,000 $$

Step-by-Step Solution

Answer

The number \(72,000,000\) in scientific notation is \(7.2 x 10^7\).

Step 1: Consider the given number

The given number is \(72,000,000\).

Step 2: Move the decimal point

Note that the decimal point is at the end of the given number. Move the decimal point 7 places to the left so that the new number is greater than or equal to 1 and less than 10. This gives \(7.2\). So, `A` is \(7.2\).

Step 3: Determine the exponent

The deciaml point was moved 7 places to the left. So `n` is \(7\).

Step 4: Write the number in scientific notation

The number in scientific notation is `A x 10^n` which is \(7.2 x 10^7\).

Key Concepts

Decimal Point

When rewriting numbers into scientific notation, the decimal point plays a crucial role. A decimal point is used in numerical figures to separate the integer part of the number from the fractional part. Imagine it as a marker that determines where 1s, 10s, and so forth, begin and end. In the original number, 72,000,000, the decimal point isn't visibly present; it's hidden at the number's end, right after all the zeros. To convert this number into scientific notation, we need to move the decimal point to a position where the resulting number is between 1 and 10. For instance, in this case, moving the decimal point seven places to the left yields 7.2. Thus:

• The integer 72,000,000 becomes the decimal 7.2.
• This shift simplifies the expression of large numbers and aids readability and comprehension.

Exponent

In scientific notation, the role of the exponent is critical. It tells us how far the decimal point has moved to convert a number into its new form. After moving the decimal point to create a number between 1 and 10, count the number of places moved to determine the exponent.If we go back to our previous example:

• By moving the decimal point seven places left to get 7.2, the exponent is 7.
• This number shows us that the original number is equivalent to our new number, multiplied by 10 raised to the power of the exponent.

The complete expression for the number is therefore written as \(7.2 \times 10^7\). The exponent encapsulates the magnitude of the original number relative to 1, making it more manageable and succinct.

Number Representation

Number representation refers to the different approaches to denote numbers numerically. Scientific notation is a powerful way, especially when dealing with very large or very small numbers. It streamlines complex figures by reducing them into simpler, comprehensible parts. Here's how it works:

• Identify the significant digits of the number. In this case: 72,000,000 transforms into 7.2. This is the 'A' part, also known as the coefficient in scientific notation.
• Next is the power part, or "exponential" part: 10 raised to some power 'n'. Here, that power is 7, because the decimal point was moved 7 times.

Together, these form the scientific notation \(7.2 \times 10^7\). This format is universally understood, especially in scientific and mathematical applications, making it a standard tool to convey complex numbers efficiently.

Mathematics Education

Scientific notation is essential in mathematics education. Understanding how to convert numbers into and out of this form enhances numerical literacy and problem-solving skills. It reduces the intimidation factor of large numbers and promotes analytical thinking. In the classroom, students learn:

• The practical steps involved in moving a decimal point, identifying a coefficient, and determining an exponent.
• How exponentiation in scientific notation enables calculations with larger numbers that would otherwise be cumbersome.

By routinely practicing these conversions, students develop comfort with mathematical expressions they will encounter in advanced subjects like physics, engineering, and technology. This builds a strong foundation for interpreting scientific data and employing it in various real-world scenarios.