Problem 129 - Elementary and Intermediate Algebra

Question

Using each of the numbers \(2,3,\) and 4 only once, what is the greatest value that the following expression can have?

Step-by-Step Solution

Answer

The greatest value is 12, using 3 and 4 in the expression as \( 3 \times 4 \).

Step 1: Understanding the Expression

The expression we need to maximize is the product of two terms, and we'll use the numbers 2, 3, and 4 only once to fill the variables. This means we'll be solving an equation of the form: \( a \times b \).

Step 2: Identifying Possible Combinations

List out all possible combinations for using the numbers 2, 3, and 4 in the expression \( a \times b \). You can pair them as (2, 3), (2, 4), or (3, 4). The number not chosen will be used in multiplication.

Step 3: Calculating Each Combination

- For (2, 3) as numbers in \( a \times b \), we calculate: \( 2 \times 3 = 6 \).- For (2, 4) as numbers in \( a \times b \), we calculate: \( 2 \times 4 = 8 \).- For (3, 4) as numbers in \( a \times b \), we calculate: \( 3 \times 4 = 12 \).

Step 4: Selecting the Greatest Value

Compare all results from the calculations in Step 3: 6, 8, and 12. The greatest value from these calculations is 12.

Key Concepts

Combination of Factors

When trying to maximize an algebraic expression like the product of two numbers, we often look at different combinations of available numbers or factors. In this problem, the numbers we can use are 2, 3, and 4. We need to find a pairing, or combination, such that their product is as large as possible.

Consider this process as building puzzle pieces together. You select two numbers as pieces and see what happens when you "put them together" through multiplication. These pieces, or combinations, are chosen to explore all the possibilities of maximizing the equation.

• One combination of factors is to choose 2 and 3.
• Another is to choose 2 and 4.
• The last one is to combine 3 and 4.

Understanding that each combination must use the numbers only once stresses the importance of planning out different scenarios before calculating the product.

Multiplying Integers

Multiplication of integers is a key concept when maximizing expressions. It involves combining two integers to get an outcome that represents how they "add up" over a certain amount of units. You'll often see equations like this: \[ a \times b \].

A neat aspect of multiplying numbers is that you can very quickly figure out large or small outcomes with just a little bit of math. With numbers like 2, 3, and 4, you just need to do the simple multiplication operation for each combination available.

• Multiply 2 by 3 to get 6.
• Multiply 2 by 4 to get 8.
• Multiply 3 by 4 to get 12.

Notice how different combinations give you different results. Integer multiplication helps us see the effects of these combinations.

Evaluating Expressions

Evaluation means calculating an expression's actual value once you've picked your combination of factors. You use operations, like multiplication, on the factors you've placed together to "solve" the expression. It essentially gives you a numerical value for each scenario you foresee.

In our problem, this involved testing each pairing of numbers through multiplication and checking which gave the biggest product. By evaluating the expression for each pair, you find the one that works best.

• The product of 2 and 3 evaluates to 6.
• The product of 2 and 4 evaluates to 8.
• The product of 3 and 4 evaluates to 12.

The act of comparing all these outcomes lets us pinpoint the greatest value, which is fundamental in maximizing expressions.