Problem 51 - Prealgebra

Question

Simplify to make the calculation as easy as possible. \(72(38)+(-72)(18)\)

Step-by-Step Solution

Answer

The simplified calculation is 1440.

Step 1: Apply the Distribution Law

Recognize that both terms share a common factor, 72. We can use the distribution law here, factoring out 72. So the expression becomes: \( 72(38) + (-72)(18) = 72(38 - 18) \).

Step 2: Simplify Inside the Parentheses

Now simplify the expression inside the parentheses: \(38 - 18 = 20\). So, the expression becomes \(72 \times 20\).

Step 3: Multiply the Simplified Terms

Finally, multiply the two simplified numbers: \(72 \times 20\). This multiplication results in \(1440\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in pre-algebra. It allows us to simplify expressions and make calculations easier. The property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. This can be expressed with the formula \(a(b+c) = ab + ac\).

In the exercise provided, we have \(72(38) + (-72)(18)\). Both terms share the common factor 72, allowing us to apply the distributive property by factoring out the common factor: \(72(38) + (-72)(18) = 72(38 - 18)\). This simplification helps to make the calculation easier, as we no longer need to deal with two separate multiplications. Instead, we simplify the expression inside the parentheses and perform one multiplication.

Factoring

Factoring is the process of breaking down an expression into simpler, multiplied components.

When we factor in mathematics, we look for numbers or variables that every term in the expression shares. Factoring is particularly helpful when multiple terms can be organized into a single term multiplied by a common factor, reducing the complexity of the expression.

In the example \(72(38) + (-72)(18)\), both components of the expression can be factored by 72. This means we can rewrite the expression as one single multiplication operation by having \(72\) outside the bracket: \(72(38 - 18)\). By factoring out the common number, we have drastically cut down the complexity of our calculation.

Simplification

Simplification in mathematics involves rewriting an expression in a more compact, easily understandable, or manageable form. It is particularly handy for reducing the calculation steps and making problems less cumbersome.

After factoring out 72 in our problem, only 38 - 18 is left to calculate. Performing the operation inside the parentheses, we get 20. So, the expression simplifies from its original form to \(72 \times 20\). Simplification often reveals the underlying simplicity of a problem that may initially seem complex.

Multiplication

Multiplication is a basic arithmetic operation used in various mathematical concepts to repeat addition of the same number. It is critical in problems involving large numbers or multiple operations.

In the simplification process, once the expression has been reduced to \(72 \times 20\), we perform multiplication. \(72 \times 20 = 1440\). This operation involves combining two numbers to form a product, in this case, 1440, which is the solution to the original expression.

Breaking down the multiplication into smaller basic operations can sometimes be useful for easier calculation, especially in mental math scenarios or large number multiplications.

Integers

Integers are whole numbers without fractions or decimals, and they include positive numbers, negative numbers, and zero. In mathematical expressions and equations, integers play a significant role as they form the basis of arithmetic operations.

Our given problem, \(72(38) + (-72)(18)\), involves integers such as 72, 38, and -18. These are operated upon through various mathematical processes like distribution, factoring, and multiplication. Understanding integers and their behavior, especially negative and positive interactions, simplifies working with them in equations.

Integers are foundational in algebra and other higher mathematics; recognizing their properties and relationships is vital for efficient computation.