Problem 19 - Intermediate Algebra

Question

For Problems \(19-32\), write each of the following in ordinary decimal notation. (Objective 2 ) For example, $$ (3.18)\left(10^{2}\right)=318 $$ $$ (2.3)\left(10^{1}\right) $$

Step-by-Step Solution

Answer

The ordinary decimal notation is 23.

Step 1: Understand the Problem

The problem requires converting a number expressed in decimal notation with a power of ten into its ordinary decimal form. We need to multiply the number 2.3 by 10 to the power of 1.

Step 2: Identify Components

Identify the components in the expression. Here, 2.3 is the base number, and it is multiplied by \(10^{1}\).

Step 3: Calculate the Power of Ten

Calculate \(10^{1}\), which equals 10. This means any number multiplied by \(10^{1}\) is multiplied by 10.

Step 4: Perform the Multiplication

Multiply the base number, 2.3, by 10: \(2.3 \times 10 = 23.0\).

Step 5: Write in Ordinary Decimal Notation

The result, 23.0, is the number in ordinary decimal notation. You can express it simply as 23, since trailing zeros after a decimal point do not change the value.

Key Concepts

Powers of Ten

Understanding the "Powers of Ten" is crucial in converting expressions into decimal notation. The power of ten is represented by an exponent, like in the expression \(10^{n}\). Here, \(n\) represents how many times 10 should be multiplied by itself.
For instance, in the expression \(10^{1}\), "1" is the exponent indicating that 10 is used once manually: \(10^1 = 10\).
This method of using powers allows us to easily manipulate numbers, making calculations quicker and more intuitive:

• \(10^{0} = 1\) (Anything to the power of zero is always 1)
• \(10^{2} = 100\)
• \(10^{3} = 1000\) and so forth.

Recognizing these basic powers of ten helps to simplify multiplication and division of numbers, especially when dealing with large or very small quantities. This understanding is the foundation for scientific and decimal notations.

Scientific Notation

"Scientific Notation" is a method for expressing very large or very small numbers in a compact form. It uses powers of ten to simplify numbers, enabling scientists and mathematicians to easily write and communicate them. A number in scientific notation takes the form \(a \times 10^{n}\), where \(1 \leq |a| < 10\) and \(n\) is an integer.
For example, the number 2.3 can be simplified to \(2.3 \times 10^{1}\) in scientific notation. This expresses how the number 2.3 is multiplied by the power of ten corresponding to the value 10. This is more efficient to use than writing out all zeroes in extended form.
This notation is particularly useful for:

• Representing numbers that are extraordinarily large, like the distance between celestial objects.
• Dealing with minute quantities, such as the size of an atom.

Thus, scientific notation is an essential tool that minimizes errors and enhances clarity in calculations.

Multiplication in Algebra

"Multiplication in Algebra" involves multiplying variables, numbers, and expressions systematically. In algebra, the multiplication of numbers like 2.3 by the power of ten is straightforward.
When you encounter an expression such as \((2.3) \times (10^{1})\), you follow these steps:

• Identify the coefficients or base numbers: Here, it's 2.3.
• Resolve the exponent into its numeral equivalent: \(10^{1} = 10\).
• Multiply these simplified numbers: \(2.3 \times 10 = 23.0\).

This operation rewrites the original expression into ordinary decimal notation. The end value, 23.0, can be often written simply as 23, as trailing zeros are not needed for whole numbers.
This process is applied to ensure that expressions are in their simplest forms before applying further operations or analyses. Understanding how to carry out multiplication effectively is foundational for engaging with more complex algebraic expressions.