Problem 84 - Prealgebra

Question

Multiply. $$2^{2} \cdot 3^{3}$$

Step-by-Step Solution

Answer

\(2^2 \cdot 3^3 = 108\).

Step 1: Understand Exponents

Recall that exponents indicate how many times the base is multiplied by itself. For instance, \(2^2\) means \(2\) multiplied by itself, or \(2 \times 2\). Similarly, \(3^3\) means \(3\) multiplied by itself three times, or \(3 \times 3 \times 3\).

Step 2: Calculate Each Exponent Separately

Begin by calculating \(2^2\) and \(3^3\). First, calculate \(2^2 = 2 \times 2 = 4\). Next, calculate \(3^3 = 3 \times 3 \times 3 = 27\).

Step 3: Multiply the Results Together

Now that you have the results for each exponent, multiply them together. Therefore, you multiply \(4\) (from \(2^2\)) by \(27\) (from \(3^3\)). The calculation is \(4 \times 27 = 108\).

Key Concepts

Understanding Multiplication

Multiplication is a fundamental arithmetic operation that forms the basis for more advanced mathematical concepts. In essence, multiplication is a shortcut for repeated addition. When we multiply two numbers, we are adding one of the numbers to itself, as many times as the other number tells us. For example, when we multiply 4 by 3, it's the same as adding 4 three times: 4 + 4 + 4.

In terms of the problem we are dealing with—multiplying powers like \(2^2 \cdot 3^3\)—the multiplication operation is applied to the results of each power calculated separately. Once you find the value for each individual power, you can then multiply these results together. This two-step process highlights the role of multiplication as a connecting operation between different powers.

Exploring Powers in Mathematics

A power is a mathematical expression that describes repeated multiplication of a number by itself. It's a way to easily write and calculate large multiplications without listing all the multiplications. For example, \(2^2\) is a power where 2 is multiplied by itself, and \(3^3\) is a bigger power where 3 is multiplied by itself three times. This is efficient for concise formulation and calculation.

Powers help simplify complex mathematical problems. They're used extensively in various fields like engineering, finance, and computer science to model exponential growth or to reduce complex multiplications to more manageable forms. Learning to work with powers enhances problem-solving skills in mathematics education.

The Importance of Mathematics Education

Mathematics education lays the foundation for logical thinking and problem-solving skills. Understanding concepts like multiplication and powers is crucial, as they are widely applied in daily life, science, and technology.

Mathematics education aims to help students develop analytical skills and the ability to interpret the world quantitatively. By mastering concepts such as multiplication of powers, students gain confidence and the ability to tackle more complex mathematical problems. These skills are essential for various academic subjects and practical aspects of life, including budgeting and scientific research.