Problem 17 - Algebra 1

Question

DECIMAL FORM Rewrite in decimal form. $$ 98 \times 10^{-2} $$

Step-by-Step Solution

Answer

The number \(98 \times 10^{-2}\) in decimal form is 0.98.

Step 1: Identify the exponent

Recognize that the negative exponent means that the decimal point should move to the left. An exponent of -2 indicates that the decimal place should shift two places to the left.

Step 2: Move the decimal place

Begin with the number 98. Since the exponent is -2, move the decimal point two places to the left. You will get 0.98.

Key Concepts

Negative Exponents

When you come across a negative exponent, it might seem a bit confusing at first. However, it's quite straightforward once you get the hang of it. Essentially, a negative exponent indicates the reciprocal of the base raised to the equivalent positive exponent. In simpler terms, it's about division rather than multiplication.
For example, if you have \(10^{-2}\), this is the same as saying \(\frac{1}{10^2}\), or \(\frac{1}{100}\). When negative exponents appear in calculations, they frequently instruct us to move the decimal point. Specifically, each negative power of 10 instructs us to move the decimal point one place to the left. A useful rule of thumb is:

• **Negative exponent**: Move the decimal to the left.
• **Positive exponent**: Move the decimal to the right.

This mental shift is essential for converting expressions with negative exponents into decimal form.

Decimal Point

Understanding the decimal point's role is vital in handling numbers, especially when converting from scientific notation or dealing with exponents. The decimal point separates the whole number part from the fractional part of a number. It plays a crucial role in determining the value of digits in a number.
When you're asked to "move the decimal point," it's crucial to visualize that action. For our example, starting with 98 and a negative exponent of -2, we moved the decimal two places to the left, transforming it into 0.98. This is like turning 98 into \(98.0\) first, then making it \(9.80\), and finally \(0.98\).
The decimal point's movement drastically changes a number's value. Moving left makes the number smaller, while moving right increases its value. Understanding this shift is foundational for math, especially when dealing with various notations and conversions.

Scientific Notation

Scientific notation is a method used to handle very large or very small numbers efficiently. It's a format where numbers are expressed as a product of a number between 1 and 10 and a power of 10. This notation simplifies calculations and clarifies understanding, especially in scientific and engineering contexts.
For instance, the number 9,800 can be expressed in scientific notation as \(9.8 \times 10^3\). The concept of scientific notation directly ties into understanding exponents, as they dictate the placement of the decimal point.
Using negative exponents in scientific notation helps represent small numbers. For example, \(0.00098\) can be written as \(9.8 \times 10^{-4}\).
This approach not only makes dealing with extreme numeric values manageable, but also standardizes the way they're referenced. Whether you’re adjusting a number with a negative exponent or converting into a decimal form, scientific notation ensures accuracy without the clutter of zeros.