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Chapter 2: Problem 97

Simplify each of the following. $$2^{3} \cdot 3^{2}$$

Short Answer

Expert verified
The simplified result of \(2^3 \cdot 3^2\) is 72.

Step by step solution

01

Evaluate the Expression Separately

First, evaluate each component of the expression separately. Calculate \(2^3\) and \(3^2\). - For \(2^3\): This means multiplying 2 by itself three times, or \(2 imes 2 imes 2\). Compute this calculation.- For \(3^2\): This means multiplying 3 by itself two times, or \(3 imes 3\). Compute this calculation.
02

Compute Each Power

Perform the calculations for each of the expressions computed individually:- \(2^3 = 2 imes 2 imes 2 = 8\)- \(3^2 = 3 imes 3 = 9\)
03

Multiply the Results

Now, multiply the results from Step 2: \(8\) and \(9\). Multiply these two numbers together to get the final simplified result: \(8 imes 9 = 72\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponents
Exponents, often referred to as powers, provide a shorthand way to express repeated multiplication of the same number. Essentially, an exponent indicates how many times a base number is multiplied by itself. For example, when you have an expression like \(2^3\), it translates to multiplying 2 three times: \(2 \times 2 \times 2\). This is much simpler and faster than writing it all out each time.

When simplifying expressions with exponents, follow these steps:
  • Identify the base (the number being multiplied).
  • Identify the exponent (the number of times the base is multiplied by itself).
  • Perform the multiplication as instructed by the exponent.
So, in our exercise, \(2^3\) results in 8 because 2 multiplied by itself three times equals 8.
Basics of Multiplication
Multiplication is one of the essential operations in mathematics, forming the foundation for more complex calculations. It's a quick way to add the same number multiple times. For instance, multiplying 3 by 2 essentially asks what is the result of adding 3 twice (3 + 3). The answer is 6.

In our example, after evaluating the exponents, you multiply the results of \(2^3\) and \(3^2\). This means you take the result of \(2^3\) which is 8, and multiply it by the result of \(3^2\) which is 9. Thus, 8 times 9 equals 72.

The multiplication follows typical rules:
  • Multiply numbers directly if they are both integers.
  • If multiplying multiple sets, perform multiplication from left to right.
  • The result gives the total as if adding the number repeatedly.
Understanding Order of Operations
The order of operations is a collection of rules that define the correct procedure to evaluate a mathematical expression. In mathematics, we have an established order which makes sure everyone simplifies equations consistently, arriving at the same correct result.

The order of operations can be remembered using the acronym PEMDAS:
  • Parentheses first
  • Exponents (i.e., powers and roots, etc.)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In our exercise, the expression \(2^3 \cdot 3^2\) involves exponents and multiplication. According to the order of operations, we handle the exponents first. Only after evaluating the powers do we proceed with the multiplication step. This ensures that the arithmetic is handled systematically and consistently.

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