Problem 103 - Intermediate Algebra

Question

3 because 8 is a perfect cube. $$ 16 $$ # Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 # 3 because 8 is a perfect cube. $$ 16 $$

Step-by-Step Solution

Answer

16 can be written as 8 # 2 because 8 is a perfect cube.

Step 1: Identify Perfect Cubes

First, identify the perfect cubes that are less than or equal to the integer 16. The perfect cubes are 1, 8, and 27, but since 27 is greater than 16, we only consider 1 and 8.

Step 2: Test Perfect Cubes as Factors

Now, test which perfect cube can be multiplied with another integer to obtain 16. Begin with 8 since it is the largest perfect cube less than 16. Check if 16 is divisible by 8.

Step 3: Perform the Division

Divide 16 by 8:\[16 \div 8 = 2\]This shows that 16 can be expressed as a product of 8 and 2.

Step 4: Confirm Factorization

Since 8 is a perfect cube and the division confirmed that it multiplied by 2 equals 16, we can conclude that 16 can be expressed as the product of 8 and 2.

Key Concepts

Integer Factorization

Integer factorization involves breaking down numbers into their multiplicative components. In this context, you're tasked with finding factor pairs where one part is a perfect cube.
When working with integers, it's essential to understand which numbers factorize into others easily. Identifying factor pairs, especially when one needs to be a perfect cube, involves checking divisibility.

• Start by determining all the perfect cubes less than the given integer. For 16, these are 1 and 8.
• Choose the largest perfect cube and check if the integer divides evenly by this cube.

This process allows breaking an integer into components, facilitating different mathematical operations and problem-solving applications.

Algebraic Expressions

An algebraic expression combines numbers and variables using operators like addition, subtraction, multiplication, and division. Understanding how to work with these expressions is crucial for manipulating and simplifying them.
In exercises like these, we're often expressing numbers in the same way we'd express algebraic terms.

• For instance, expressing 16 as a product of 8 and 2 mimics expressing a complex term in its simplest form by breaking it down into smaller components.
• This simplification helps in identifying patterns and relationships between different mathematical entities, crucial for solving equations.

Simplification and rewriting expressions are key skills in algebra, aiding in the deeper understanding of how numbers interact in an equation.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to solve problems and understand concepts. In this instance, it's about strategically choosing numbers that make solving the problem efficient and clear.
The reasoning process for factorization involves:

• Identifying suitable perfect cubes without overlooking or skipping possible solutions.
• Logically testing factor compatibility by checking divisibility, ensuring no miscalculations.

Exercising clear reasoning during such tasks aids not only in solving the current problem but also builds skills applicable to more advanced mathematical concepts. This logical approach teaches students to approach problems methodically, a valuable trait in both theoretical and practical applications of mathematics.