Problem 71 - Algebra 1: Concepts and Skills

Question

WRITING POWERS Write the expression in exponential form. (Lesson \(1.2)\) \(9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9\)

Step-by-Step Solution

Answer

The exponential form of the expression \(9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9\) is \(9^6\).

Step 1: Identify the base and count the repetitions

The base is the number that is being multiplied. In the provided expression, the base is number 9. Count how many times number 9 appears in the multiplication. There are 6 instances of number 9.

Step 2: Write in Exponential Form

Having known that number 9 is the base and it appears 6 times, now write in exponential form as: \(9^6\) . Number 9 is the base and 6 is the exponent, which means 9 is multiplied by itself 6 times.

Key Concepts

Writing Powers

When we talk about writing powers, we're essentially trying to simplify expressions that involve multiplying the same number over and over again. Instead of writing the number multiple times, we use a single, concise representation called the exponential form. For example, the multiplication expression \(9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9\) can be written in power form as \(9^6\). The key benefit of this form is that it saves space and makes large computations easier to understand and manage.

Using powers is an efficient way to handle repeated multiplication because:

• It reduces complexity and keeps work neat.
• It simplifies the process of working with large numbers.
• It is a fundamental concept in higher-level math, crucial for algebra, calculus, and beyond.

Understanding how to write powers is essential for solving many math problems quickly and efficiently.

Base and Exponent

In exponential forms like \(9^6\), two components stand out: the base and the exponent. These are key elements in understanding and using powers effectively. Let's break down what these terms mean:

• Base: The base is the number that you are multiplying by itself. In our example, the base is 9. This is the number you'll see repeated in the multiplication expression.
• Exponent: The exponent tells you how many times the base multiplies itself. In the example \(9^6\), the exponent is 6. This indicates that the base 9 is used in a multiplication chain 6 times.

The relationship between the base and the exponent is what forms the 'power'. It's essential because it clearly and concisely defines how many times you multiply the base by itself. Knowing how to identify these in any multiplication problem is foundational for building your math skills.

Multiplication Repetition

The concept of multiplication repetition occurs frequently in mathematics. It's all about recognizing patterns where the same number is multiplied repeatedly. This repetitive action is what lays the groundwork for using powers like we did in expressing \(9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9\) as \(9^6\).

In everyday scenarios, you encounter multiplication of this nature often, such as calculating compound interest, geometric growth, or computer processing operations. Recognizing multiplication repetition is helpful because:

• It underlines the power of doing the same action multiple times, enhancing calculation efficiency.
• It helps in understanding proportional growth and scaling.
• It forms the basis for comprehending more advanced mathematical concepts such as functions and series.

By mastering this concept, you open the door to solving complex problems with ease.