Problem 18 - Fundamentals of Mathematics

Question

Write the expressions using exponential notation. $$10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10$$

Step-by-Step Solution

Answer

The expression is \(10^6\).

Step 1: Identify the Base

The expression given is composed of repeated factors of 10. Therefore, the base in this expression is 10.

Step 2: Count the Number of Factors

Count how many times the base (10) is multiplied by itself. In this case, there are 6 factors of 10 in the expression.

Step 3: Write the Exponential Expression

Using the base and the exponent (number of times the base is multiplied by itself), represent the expression using exponential notation. Thus, the expression is written as \(10^6\).

Key Concepts

Base and Exponent

In exponential notation, we often encounter two key components: the **base** and the **exponent**. These elements simplify the expression of numbers that are multiplied repeatedly. The **base** is the number that is being multiplied. In the expression you provided, that number is 10. The **exponent**, on the other hand, tells us how many times the base is used as a factor in the expression. For example:

• If the expression is 10 multiplied by itself six times, the exponent is 6.
• This is shown as a small number written to the upper right of the base, like this: \(10^6\).

Using exponential notation makes it easier to write and understand very large or very small numbers, enhancing both readability and mathematical efficiency.

Repeated Multiplication

When we discuss repeated multiplication, we refer to a scenario where the same number is multiplied by itself multiple times. It's a concept that exponential notation captures in a compact way.Consider the given expression:

• 10 multiplied by itself six times can be visualized as: \(10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10\).
• Count these factors carefully to determine the exponent, which in this case is 6.

This concept is fundamental because it allows us to shift from lengthy expressions to concise notation, facilitating simpler mathematical operations and calculations.

Mathematics Expression Simplification

Simplifying mathematical expressions is a crucial skill, particularly when dealing with multiplication. By transitioning to exponential notation, lengthy multiplication problems can be reduced to a couple of elements.Why simplify?

• It makes calculations quicker and easier, especially with larger numbers or more complex equations.
• Helps in avoiding errors when performing manual calculations by reducing the repetition of writing the base multiple times.
• Let's not forget, it is essential for interpreting and solving scientific and engineering problems where such notation is prevalent.

In our example, instead of repeatedly writing the base, we just write \(10^6\), which instantly tells us the repeated multiplication and simplifies the entire mathematical process.