Problem 19 - Algebra 1: Concepts and Skills

Question

Tell whether you would move the decimal left or right and how many places to write the number in decimal form. $$ 9.04 \times 10^{-7} $$

Step-by-Step Solution

Answer

The decimal form of \(9.04 \times 10^{-7}\) is \(0.000000904\). The decimal point is moved 7 places to the left.

Step 1: Understand the Scientific Notation

In scientific notation, the number \(9.04 \times 10^{-7}\) has a decimal and an exponent part. A negative exponent indicates that the decimal has to be moved to the left.

Step 2: Move the Decimal Point

The decimal point in \(9.04\) has to be moved \(7\) places to the left because the exponent of \(10\) is \(-7\). This means you add six zeros to the left of \(9.04\) and shift the decimal point after the first zero.

Step 3: Write the Decimal Form

After moving the decimal point seven places to the left, \(9.04 \times 10^{-7}\) becomes \(0.000000904\) in decimal form.

Key Concepts

Decimal Point Movement

When you're dealing with scientific notation, the location of the decimal point plays a crucial role. Understanding how to move this point can simplify converting scientific notation to a more familiar decimal form.

In simple terms, the exponent in scientific notation indicates how many places the decimal should be moved. If the exponent is positive, move the decimal to the right, which often results in a larger number. Conversely, a negative exponent means you move the decimal to the left. This typically results in a smaller number, as the value becomes a fraction of one.

To illustrate, consider moving the decimal in the number for the given exercise: \(9.04 \times 10^{-7}\). Since the exponent is \(-7\), you shift the decimal point 7 places to the left. You'll need to add zeros to fill the spaces if there aren't enough digits. In this case, you'll add six zeros to make it move seven places.

Negative Exponent

The concept of a negative exponent in scientific notation is like giving you directions about where to move the decimal: backwards or to the left. This negative value doesn't mean the number is negative, but rather, it's about the scaling of the original number.

For instance, the equation \(9.04 \times 10^{-7}\) indicates the decimal should be repositioned to the left, condensing the number to a very tiny fraction of its original size. This is especially useful in fields like physics and chemistry, where scientists work with extremely small measurements.

To process \(9.04 \times 10^{-7}\), interpret the \(-7\) as a guide for placing the decimal point leftwards by 7 spaces. This doesn't affect the central value \(9.04\) but rather alters the scale, expressing it as a much smaller decimal.

Converting to Decimal Form

Converting a number from scientific notation to decimal form involves repositioning the decimal point based on the exponent. This transformation makes it easier to visualize and comprehend, especially for everyday calculations.

Once you determine the movement based on the negative exponent, rewrite the number in its new position. For example, when shifting the decimal for \(9.04 \times 10^{-7}\):

• Identify the current position of the decimal in \(9.04\) (which is right after the initial digit \(9\)).
• Recognize that the \(-7\) suggests moving the decimal 7 places left. You add zeros accordingly to accommodate this move.

For this example, write zeros in front of \(9.04\) to cover the needed places; finally, position the decimal, resulting in \(0.000000904\).

By practicing this method, converting numbers between these forms can become second nature, ensuring you can switch efficiently based on the context of the problem.