Problem 2 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(10^{3}, 10^{50},\) and \(10^{-4}\) are ________ of 10

Step-by-Step Solution

Answer

Powers

Step 1: Understanding Powers of 10

To solve the problem, let's first clarify what a 'power of 10' means. A power of 10 is a number that can be expressed as the exponential expression of 10 raised to an integer. The expression uses the form \(10^n\), where \(n\) is the exponent, which can be a positive number, a negative number, or zero.

Step 2: Analyze Given Numbers

Next, review each number given in the exercise: \(10^3\), \(10^{50}\), and \(10^{-4}\). Each of these numbers can be expressed in the form of an exponent of 10, with corresponding powers 3, 50, and -4, respectively. This aligns with the definition of a power of 10.

Step 3: Fill in the Blank

Now that we understand the concept, we can conclude the type of numbers \(10^3\), \(10^{50}\), and \(10^{-4}\) represent. They are all powers of the base number 10.

Key Concepts

Understanding the Exponent

In mathematics, an **exponent** is a fundamental concept that refers to the number that indicates how many times the base number is multiplied by itself. For example, in the expression \(10^3\), the exponent is 3. This means that 10 is multiplied by itself three times: \(10 \times 10 \times 10 = 1,000\).

• The exponent is written as a small number to the top right of the base number.
• Positive exponents signify repeated multiplication.
• Negative exponents indicate division, essentially representing the reciprocal.
• An exponent of zero means that the base number is raised to zero power, which always equals 1.

When you understand the role of the exponent, you can manage numbers more efficiently, especially when dealing with scientific notation or engineering fields.

Exploring the Base Number

The **base number** in an exponential expression is the number that is raised to a power. Taking the expression \(10^3\) as an example, 10 is the base number.

• The base is the quantity you begin with before applying the exponent's instruction of how many times to multiply or divide.
• In most exponential expressions concerning powers of 10, the base is commonly 10, but it can be any real number.
• Knowing the base allows you to understand what the 'unit' is that gets repeatedly multiplied or divided.

Base numbers are crucial in understanding exponential growth or decay. They serve as the starting point for further calculations and transformations.

Deciphering Exponential Expressions

An **exponential expression** is a mathematical notation used to describe expressions that involve powers of numbers. It combines both the base and the exponent neatly into one expression.

• Consider the expression \(10^{50}\). Here, it's an exponential expression where 10 is the base and 50 is the exponent.
• Exponential expressions are compact and efficient for representing very large or very small numbers.
• These expressions are ubiquitous in scientific and technical disciplines because they simplify complex calculations.

Exponential expressions provide a powerful tool for conveying the concept of growth, scaling, and changes over various magnitudes. Understanding them helps in interpreting scientific data, economic trends, and more. Mastery of using and manipulating exponential expressions can significantly enhance your mathematical prowess.