Problem 71 - Algebra and Trigonometry

Question

Write each number in decimal notation without the use of exponents. $$ 7.9 \times 10^{-1} $$

Step-by-Step Solution

Answer

The decimal notation of \(7.9 \times 10^{-1}\) is 0.79.

Step 1: Identify the Exponent Value

In the given exercise, the base number is 7.9 and it is multiplied by 10 to the power of -1, meaning \(10^{-1}\). This negative exponent tells us that the decimal point needs to be moved one place to the left.

Step 2: Move the Decimal Point

Here, the base number is 7.9 and the decimal is between 7 and 9. Moving this decimal one place to the left, it will be in front of 7.

Step 3: Write Number In Decimal Notation

The number written in decimal notation after moving the decimal point one place to the left from 7.9 is 0.79.

Key Concepts

Negative Exponent

In mathematical notation, exponents are used to indicate how many times a number, called the base, is multiplied by itself. When we see a negative exponent, it signifies a different operation. A negative exponent represents division or taking the reciprocal. For example, when we have a negative exponent like \(10^{-1}\), it instructs us to take the reciprocal of \(10\).

So, \(10^{-1}\) is equal to \(\frac{1}{10}\). This is why negative exponents cause the decimal point to move, helping in representing fractions more conveniently. Getting comfortable with the concept of negative exponents will allow you to easily switch between fractional and exponent forms.

Move the Decimal Point

When you're dealing with powers of ten, moving the decimal point is the visual step that translates exponent rules into decimal notation.

For example, in the number \(7.9 \times 10^{-1}\), the exponent \(-1\) directs us to shift the decimal point one place to the left. Here's how it works:

• Start with the decimal point at its original position between 7 and 9 in \(7.9\).
• The exponent of \(-1\) tells us to move the decimal point one place leftward.
• After moving, the new number becomes \(0.79\).

This simple movement makes handling and understanding numbers in scientific notation much easier.

Scientific Notation

Scientific notation is a powerful way of expressing very large or very small numbers succinctly and clearly. It simplifies complex calculations and allows easy comparison of different magnitudes.

In scientific notation, a number is written as the product of two parts:

• A decimal number (called the significand) that is usually between 1 and 10.
• A power of 10 that indicates the number of places the decimal point is moved.

For instance, \(7.9 \times 10^{-1}\) shows that the decimal needs to be moved one place to the left, converting it into \(0.79\). Understanding scientific notation provides a crucial tool for solving problems involving varying scales. By mastering this notation, you can efficiently handle very large or very small quantities with ease.