Problem 32 - Algebra 1

Question

SCIENTIFIC NOTATION Rewrite in scientific notation. $$ 2,730,000,000 $$

Step-by-Step Solution

Answer

The number 2,730,000,000 in scientific notation is \(2.73 x 10^9\).

Step 1: Identify the digit term

First, find the first term \(a\) which is simply the given number, removing trailing zeros and adding a decimal after the first digit. For this number, \(a\) is 2.73.

Step 2: Determine the exponent

Next, we find the exponent \(n\). This is the number of places we moved the decimal point. Count the number of places moved from the end of original number up to where it is now in \(a\). The number of places moved will be \(n\). In this case, we moved the decimal point 9 places to the left, so \(n\) is 9.

Step 3: Write the number in scientific notation

Now that we have values for \(a\) and \(n\), we put it all together to write the number in scientific notation as \(a x 10^n\). Thus the number 2,730,000,000 in scientific notation will be \(2.73 x 10^9\).

Key Concepts

Exponent

When working with scientific notation, the exponent plays a crucial role. It determines how many times you multiply the base number, which is generally 10, to convert the given number back to its original form. In our example, the base is implicitly 10. The exponent tells us that the decimal point has been moved a certain number of places. If the exponent is positive, as in our case with 9, the decimal moves to the right. If it were negative, the decimal would go to the left.Understanding this concept makes converting large or very small numbers much simpler. It eliminates the need for writing out cumbersome zeros. So, when we see an exponent of 9 in the expression \(2.73 \times 10^9\), it signifies that the decimal needs to move 9 places to the right to go from 2.73 to 2,730,000,000.

• Positive exponent: Decimal shifts right.
• Negative exponent: Decimal shifts left.
• Zero exponent: Decimal stays put (e.g., \(10^0 = 1\)).

This makes scientific notation a powerful tool for managing and simplifying very large or very small numbers.

Decimal Point

The decimal point is pivotal when converting numbers into scientific notation. It's all about making sure that we have just one non-zero digit to the left of the decimal point in the number part of the notation. In the initial step of our exercise, we take the large number 2,730,000,000 and focus on the digits that define its value. We shift the decimal point from the end to just after the first significant digit: 2.73. This step ensures that the number is presented in a form that's concise and easy to handle.

• The repositioning of the decimal maximizes readability.
• Results in a simpler, cleaner representation of the number.

Remember, wherever the decimal point ends up in the process of moving, that's where the true value of the number is concentrated. This maneuver reduces the potential for error when reading or calculating with large numbers.

Trailing Zeros

Trailing zeros are the zeros that come after the last non-zero digit in a whole number. In scientific notation, trailing zeros in the original number play a big role in determining the size of the exponent.Take our example, 2,730,000,000, and observe how many zeros trail at the end. By removing these trailing zeros and moving the decimal point before just the first non-zero digit, you demonstrate the full magnitude of the number using the exponent.

• Trailing zeros help ascertain the magnitude of the exponent.
• They don't contribute to the value once transformed into scientific notation.

For instance, with 9 trailing zeros, the scientific notation \(2.73 \times 10^9\) captures the essence of our original number without the clutter of zeros. This methodology maintains efficiency and accuracy in calculations, especially when dealing with particularly immense or minuscule figures.