Problem 67 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 3\left(2 a+2 a^{2}\right)+8\left(3 a+3 a^{2}\right) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: $3(2a + 2a^2) + 8(3a + 3a^2)$ Answer: $30a + 30a^2$

Step 1: Apply the distributive property

To simplify the expression, we need to apply the distributive property for both terms: $$ 3\left(2 a+2 a^{2}\right)+8\left(3 a+3 a^{2}\right) \\ 3(2a) + 3(2a^2) + 8(3a) + 8(3a^2) $$

Step 2: Multiply the numbers

Now, multiply the numbers for each term: $$ 6a + 6a^2 + 24a + 24a^2 $$

Step 3: Combine like terms

Finally, combine the like terms (terms with the same variable and exponent): $$ (6a + 24a) + (6a^2 + 24a^2)\\ 30a + 30a^2 $$ The simplified algebraic expression is: $$ 30a + 30a^2 $$

Key Concepts

Distributive Property

Understanding the distributive property is essential when simplifying algebraic expressions. It allows us to multiply a single term by a group of terms within a parenthesis. The general formula is \( a(b+c) = ab + ac \). When we apply this property, each term within the parenthesis is multiplied by the term outside of it.

Let's look at the given exercise as an example. The expression \( 3(2a+2a^2) + 8(3a+3a^2) \) involves two sets of parentheses. By applying the distributive property, we multiply 3 by both \( 2a \) and \( 2a^2 \) separately, then similarly multiply 8 by both \( 3a \) and \( 3a^2 \) separately. This step breaks down the complex expression into a series of simpler terms that can be more easily managed in subsequent steps.

Combining Like Terms

Combining like terms is a method used to simplify an algebraic expression, making it easier to work with. Like terms are terms that have the same variables raised to the same power—only the coefficients may differ. For example, \( 2a \) and \( 3a \) are like terms since they both contain the variable 'a' to the power of 1.

After using the distributive property in our exercise, we are left with the terms \( 6a \) and \( 24a \) as well as \( 6a^2 \) and \( 24a^2 \). To combine like terms, we add up the coefficients of those terms that are alike. For instance, \( 6a + 24a = 30a \) and \( 6a^2 + 24a^2 = 30a^2 \). This creates a much simpler expression where each type of term is represented only once.

Algebraic Simplification

Algebraic simplification encompasses several processes, including using the distributive property and combining like terms, to transform a complicated expression into a simpler, more digestible form. The purpose is to make it more straightforward to work with or to solve, whether that be for evaluating the expression, solving equations, or other algebraic manipulations.

In our example, after distributing and combining like terms, we simplify \( 6a + 6a^2 + 24a + 24a^2 \) to get \( 30a + 30a^2 \). This final expression is easier to evaluate for any given value of 'a', and it showcases the effectiveness of algebraic simplification. Good practices in algebraic simplification include performing one step at a time and thoroughly checking your work for any additional like terms that can be combined or any further simplifications that can be made.

Elementary Algebra

Elementary algebra is the foundational study of algebra that includes basic algebraic techniques used for solving equations and expressions involving variables. It introduces important concepts such as variables, exponents, and the basic properties of operations like the distributive property, associativity, and commutativity.

Exercises like the one provided offer practice in these basic techniques to build proficiency. By simplifying the expression \( 3(2a+2a^2) + 8(3a+3a^2) \) using elementary algebra skills, students gain hands-on experience that reinforces their understanding of the subject. It's important to master these basic concepts and processes as they serve as building blocks for more advanced studies in algebra and other branches of mathematics.