Question

In Problems 129-136, write each number as a decimal. \(8.1 \times 10^{-2}\)

Short answer

0.081

Step-by-step solution

Identify the Number and the Exponent

First, let's identify the base number and the exponent. We have the base number as 8.1 and the exponent as -2 in the given problem: \(8.1 \times 10^{-2}\).

Understand the Exponent

A negative exponent means we will be moving the decimal point to the left. The exponent -2 tells us to move the decimal point 2 places to the left.

Move the Decimal Point

Starting with 8.1, move the decimal point 2 places to the left: \(8.1 \rightarrow 0.081\).

Key concepts

negative exponent

In mathematics, exponents are used to express repeated multiplication. When we see a negative exponent, it signifies a different action. A negative exponent indicates that the base number should be divided by the corresponding power of 10 instead of being multiplied. For example, consider the expression \(10^{-2}\). Instead of multiplying by 10 twice, we divide by 10 twice:

• First division: \(10^{-1} = \frac{1}{10} = 0.1\)
• Second division: \(10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01\)

In our given problem, the base number is 8.1 and the exponent is -2, which means we need to divide 8.1 by \(10^2\) or 100. To do this easily, we move the decimal point to the left by as many places as the exponent value. Here, we will move it 2 places to the left.

moving decimal point

Understanding how to move the decimal point is essential when dealing with exponents. Moving the decimal is a visual way to perform multiplication or division by powers of 10. In the case of a negative exponent, we move the decimal to the left.
Let's illustrate this with our example:\(8.1 \times 10^{-2}\). Starting with the decimal in 8.1, we move it 2 places to the left:
1. The first move shifts the decimal from 8.1 to .81
2. The second move shifts it from .81 to .081
This gives the result as 0.081.

scientific notation

Scientific notation is a method of writing very large or very small numbers compactly. It is expressed in the form \(a \times 10^n\), where \(a\) is a number greater than or equal to 1 and less than 10, and \(n\) is an integer. This notation makes it easier to handle arithmetic with very large or very small amounts.
For example, the number 0.00045 can be written as \(4.5 \times 10^{-4}\), and the number 3000 can be written as \(3 \times 10^3\).
In our problem, 8.1 is written in scientific notation as \(8.1 \times 10^{-2}\). This tells us that to convert it back to a decimal, we need to move the decimal point 2 places to the left, resulting in 0.081. Using scientific notation properly helps simplify calculations and makes large or small numbers more manageable.