Problem 101 - Intermediate Algebra

Question

2 because 9 is a perfect square. $$ 48 $$ # Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as 9 # 2 because 9 is a perfect square. $$ 48 $$

Step-by-Step Solution

Answer

48 can be written as 16 # 3, where 16 is a perfect square.

Step 1: Understand the Problem

We need to express the integer 48 as a product of two numbers where one of the numbers is a perfect square. A perfect square is an integer that is the square of an integer.

Step 2: Prime Factorization

Find the prime factorization of 48 to help identify perfect squares. The prime factorization of 48 is:\[ 48 = 2^4 \times 3^1 \]

Step 3: Identify Factors

From the prime factorization, identify combinations that form perfect squares. Here, \(2^4 = 16\) is a perfect square since it is \((2^2)^2 = 16\). Another perfect square from the factorization is \(4 = (2^2)\).

Step 4: Choose the Largest Perfect Square

Choose the largest perfect square possible from the factorization, which is 16. Write 48 as a product of this perfect square and another factor: \[ 48 = 16 \times 3 \]

Step 5: Write the Final Expression

Write 48 as a product of one perfect square and another integer: \[ 48 = 16 \times 3 \]Here, 16 is a perfect square (\(4^2\)).

Key Concepts

Perfect Square

A perfect square is a number that results from multiplying an integer by itself. For example, 9 is a perfect square because it is the product of 3 multiplied by 3, or \(3^2\). Recognizing perfect squares is useful in various mathematical operations, especially when simplifying or reconfiguring numbers.

The process of squaring is simple:

• Take an integer, such as 4.
• Multiply the integer by itself: \(4 \times 4 = 16\).

Thus, 16 is a perfect square, as it is the result of \(4^2\). Perfect squares are helpful in factorization tasks, as seen in exercises that involve rewriting numbers as a product of factors or identifying square roots.

In the given example, identifying the largest possible perfect square factor helps decompose the integer more efficiently. For the number 48, 16 (which is \(4^2\)) is the largest perfect square factor.

Integer Factorization

Integer factorization involves breaking down a composite number into a product of its integer factors, specifically prime numbers. Knowing how to factor integers is essential for finding the building blocks of a number and can aid in solving different types of equations and problems.

For example, consider the number 48 in the exercise. The goal is to find its prime factorization, which is essentially the number expressed as a product of prime numbers:

• 48 can be broken down to \(2^4 \times 3^1\).

This tells us that the prime factors of 48 are 2 and 3. Factorization helps identify combinations that form perfect squares, like \(2^4\), which leads to simplifying or rewriting the number in desired forms. The ability to factor integers is crucial for understanding the structure of numbers.

Multiplication of Integers

Multiplication of integers is a basic arithmetic operation that involves combining groups of equal sizes. It serves as the foundation for more complex mathematical operations and concepts. Here’s a quick refresher:

• To multiply two integers, like 4 and 3, you perform the operation \(4 \times 3 = 12\).

Multiplying integers can produce larger numbers, and when applied in factorization, it helps verify various representations of the same number. In factorization exercises, once perfect squares are identified, multiplication is used to reconstruct the original number through different combinations of factors.

For example, in the exercise, after identifying 16 as a perfect square that is a factor of 48, we multiplied it by 3 to get back to 48 (\(16 \times 3 = 48\)). Understanding multiplication also aids in verifying decomposition results and ensuring accuracy in calculations.