Problem 91 - Elementary Algebra

Question

Simplify \(3\left(5 n+6 m^{2}\right)-2\left(3 n+4 m^{2}\right)\).

Step-by-Step Solution

Answer

Question: Simplify the expression \(3\left(5n+6m^{2}\right) - 2\left(3n+4m^{2}\right)\). Answer: The simplified expression is \(9n + 10m^{2}\).

Step 1: Distribute the constants

Distribute the '3' to both terms inside the first set of parentheses and the '-2' to both terms inside the second set of parentheses: \(3\left(5n+6m^{2}\right) - 2\left(3n+4m^{2}\right) = 15n + 18m^{2} - 6n - 8m^{2}\).

Step 2: Combine like terms

Combine the terms with the same variables: \((15n-6n) + (18m^{2} - 8m^{2}) = 9n + 10m^{2}\). The simplified expression is \(9n + 10m^{2}\).

Key Concepts

Distributive Property

Let's explore the distributive property, a cornerstone of algebra that allows you to simplify expressions that include parentheses. Think of distribution as a way to eliminate the parentheses by 'distributing' the number outside the parentheses to each term inside.

For example, consider the algebraic expression: \( a(b + c) \). Using the distributive property, we would multiply 'a' with both 'b' and 'c' separately. As a formula, it can be expressed as: \( a(b + c) = ab + ac \).

In the exercise, the distributive property is applied twice: first with 3 being distributed over \(5n + 6m^2\), and then with -2 being distributed over \(3n + 4m^2\). This step is crucial because it sets the stage for combining like terms next.

Combine Like Terms

Once the distributive property has been applied, the next step is to 'combine like terms.' Like terms are terms that have the exact same variable parts, meaning they have the same variables raised to the same power. Numbers without variables, known as constants, can also be like terms with each other.

In our exercise, after distribution, we get terms with 'n' and terms with \(m^2\) as variables. To simplify, we combine the coefficients (numerical values) of these like terms. This involves basic arithmetic – adding or subtracting the coefficients: \(15n - 6n\) simplifies to \(9n\), and \(18m^2 - 8m^2\) simplifies to \(10m^2\). By combining like terms, we streamline the expression into its simplest form.

Helpful Tip:

Always look for terms with the same variable and exponent pairings – these are your 'like terms' that can be combined to simplify an expression.

Algebraic Expression Simplification

Algebraic expression simplification is the process of making an algebraic expression as straightforward as possible. This process involves multiple steps, including using the distributive property, combining like terms, and reducing fractions or simplifying exponents when possible.

The goal of simplification is to make the expression easier to understand and work with, especially for solving equations or evaluating the expression for specific variable values. As we've done in the exercise, once like terms are combined, we often find that the expression looks much less daunting and is simpler to use.

Key Takeaway:

Remember, the aim is not just to perform operations correctly, but to transform the expression into its least complicated form without changing its value.

Elementary Algebra

Elementary algebra is the foundation on which more complex mathematical concepts are built. It involves operations on algebraic expressions, solving equations, and understanding basic polynomial structures.

When working with expressions like in our exercise, you are practicing elementary algebra. It’s important to understand and follow the proper order of operations and use the properties of real numbers to manipulate expressions. Whether it's for academic purposes or for solving real-world problems, elementary algebra is a skill that underpins a vast array of disciplines.

Remember:

Mastering the basics in elementary algebra, such as recognizing like terms and correctly applying the distributive property, will make it much easier to tackle more advanced math later on.