Problem 73 - Prealgebra

Question

Factor each of the following numbers into the product of two numbers, one of which is a perfect square. (Remember from Chapter 1, a perfect square is \(1,4,9,16,25,36, \ldots\) etc. $$50$$

Step-by-Step Solution

Answer

50 = 2 * 25, where 25 is a perfect square.

Step 1: Identify Perfect Squares

First, identify the perfect squares less than 50. They are 1, 4, 9, 16, 25, and 36.

Step 2: Find Factor Pairs

Find factor pairs of 50 and check if one of the factors is a perfect square. The factor pairs are (1, 50), (2, 25), and (5, 10). Now, check for any perfect squares.

Step 3: Select the Factor Pair with a Perfect Square

From the factor pairs, (2, 25) has 25 as a perfect square. So, we can write 50 as a product of 2 and 25.

Key Concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, 4 is a perfect square because it can be written as \(2 \times 2\), and 9 is a perfect square because it equals \(3 \times 3\). Recognizing perfect squares is an essential skill in prealgebra factoring since it simplifies complex expressions and helps in mathematical problem-solving.
To identify perfect squares, look at numbers such as 1, 4, 9, 16, 25, and so on. Each of these results from multiplying a whole number by itself.

• 1 is \(1 \times 1\)
• 4 is \(2 \times 2\)
• 9 is \(3 \times 3\)
• 16 is \(4 \times 4\)
• 25 is \(5 \times 5\)

Knowing these makes it easier to simplify larger expressions or engage in factorization tasks.

Factor Pairs

Factor pairs are two numbers that multiply together to result in the original number. For example, 50 has several factor pairs, specifically (1, 50), (2, 25), and (5, 10). Understanding factor pairs allows us to break down numbers into smaller, more manageable parts, which is extremely useful in algebra and number theory.
When searching for factor pairs, list the pairs of numbers that result in the product when multiplied together. This process can also uncover potential perfect squares within the number's factors, aiding further in simplification. Here's how it's done:

• Find a number that divides evenly into your number
• Pair it with the quotient from that division
• Collect all such pairs until no new pairs can be found

Ensuring that one of your pairs contains a perfect square helps in simplifying expressions and understanding the structure of numbers.

Number Factorization

Number factorization refers to breaking down a number into its component parts or factors. The goal is to express the original number as a product of other numbers. This process can often reveal hidden perfect squares and simplify mathematical expressions.
In the context of prealgebra, learning to factor numbers like 50 involves identifying all possible factor pairs and seeing if any of those pairs contain a perfect square. For instance, once we identify the factor pairs of 50, we see that (2, 25) is a pair where 25 is a perfect square. Thus, 50 can be expressed as \(2 \times 25\).

• Begin by identifying all factor pairs
• Look for a factor pair where one number is a perfect square
• Reconstruct the original number using this pair

This method not only aids in simplification but also enhances understanding of the numerical relationship between factors, paving the way for more advanced algebraic techniques.