Problem 72 - Algebra 1

Question

\(3 x \cdot 3 x \cdot 3 x \cdot 3 x \cdot 3 x\)

Step-by-Step Solution

Answer

The result of the given multiplication written in exponential form as \(3^5\) is 243.

Step 1: Identify the Base and Exponent

The base is the number being multiplied, that is \(3\). The exponent is the number of times the base is being multiplied by itself, and from the above equation, counting instances of \(3\), the exponent is \(5\).

Step 2: Write in Exponential Form

Now that we have identified the base as \(3\) and the exponent as \(5\), this multiplication can be written in exponential form, which is \(3^5\).

Step 3: Evaluate the Exponential Expression

To calculate the result of \(3^5\) we multiply \(3\) by itself five times. Therefore, \(3^5 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243\).

Key Concepts

Base and Exponent

In mathematics, when you encounter repeated multiplication of the same number, you are dealing with exponents. And to understand exponents, it is key to grasp the idea of the base and the exponent itself. Here’s what they mean:

• Base: This is the number that is being multiplied. For instance, in the original exercise with the repeated term \(3x\), the base is the number \(3\).
  
• Exponent: This signifies how many times the base is used as a factor in the multiplication. It's like a concise form of saying "this many times". In our example, the number \(3\) is multiplied by itself 5 times, so the exponent is \(5\).
  

Now that we’ve defined these, whenever you see something like \(3^5\), you'll know it means taking the base (\(3\)) and multiplying it by itself as many times as the exponent (\(5\)) hints at. It simplifies the expression while making computations and understanding easier.

Exponential Form

Writing in exponential form turns a long multiplication like \(3 \times 3 \times 3 \times 3 \times 3\) into something sleeker and cleaner, such as \(3^5\). But why do we convert to exponential form?

• Efficiency: It's a straightforward way to show large numbers or repeated multiplications without writing them all out.
  
• Clarity: It makes the mathematical operation more understandable and concise to both write and interpret.
  

By identifying the base and counting how many it multiplies, you reduce clutter. So in this instance, simply note that \(3\) is your base and it's repeated 5 times, giving you the neat expression \(3^5\). This helps you manage complex calculations and share information with precision and clarity.

Exponential Expressions

Exponential expressions like \(3^5\) not only succinctly express repeated multiplication, but they also form the foundation for more advanced concepts in mathematics. Here’s why they matter:

• Easy Manipulation: Such expressions allow you to apply rules of exponents that simplify calculations, like multiplying or dividing powers with the same base.
  
• Scalability: Working with exponents means you can easily scale numbers up or down, which is invaluable for real-world applications in science and engineering.
  

When you evaluate an exponent like \(3^5\), you're doing what is called "raising to a power". It involves multiplying the base by itself, according to the exponent. This simplifies the original exercise from a string of multiplications to finding a single precise number, which in this case, is \(243\). Embracing exponential expressions helps you in both understanding and solving larger mathematical structures with ease.