Problem 42 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ -9 w^{5}-9 w^{4}-9 w^{5}+10 w^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(-9 w^{5} - 9 w^{4} - 9 w^{5} + 10 w^{4}\). Answer: The simplified algebraic expression is \(-18w^5 + w^4\).

Step 1: Identify like terms

In the given expression, $$ -9 w^{5} - 9 w^{4} - 9 w^{5} + 10 w^{4}, $$ we have two sets of like terms: 1. Terms with \(w^5\): \(-9w^5\) and \(-9w^5\) 2. Terms with \(w^4\): \(-9w^4\) and \(10w^4\)

Step 2: Combine like terms

Now we will combine the like terms by adding or subtracting them. For terms with \(w^5\), $$ -9 w^{5} - 9w^{5} = (-9 - 9)w^5 = -18w^5 $$ For terms with \(w^4\), $$ -9 w^{4} + 10 w^{4} = (-9 + 10)w^4 = 1w^4 $$

Step 3: Write the simplified expression

Combine the simplified terms with \(w^5\) and \(w^4\) to get the final simplified expression: $$ -18w^5 + 1w^4 $$ So, the simplified algebraic expression is: $$ -18w^5 + w^4 $$

Key Concepts

Combining Like Terms

Understanding how to combine like terms is a fundamental skill in algebra. Like terms are terms that have the exact same variables raised to the same power. For example, in the expression \( -9 w^{5} - 9 w^{4} - 9 w^{5} + 10 w^{4} \), there are two types of like terms: those with \( w^5 \) and those with \( w^4 \) coefficients. These like terms can be added or subtracted from one another because they represent the same quantities, just in different amounts.

When you combine like terms, you effectively simplify the expression by reducing the total number of terms. This is done by adding the coefficients of like terms. So, for the terms with \( w^5 \) you have \( -9w^5 \) and \( -9w^5 \), which gives \( -18w^5 \). To visualize this, imagine you have -9 apples with a special sticker (\( w^5 \) here) and you find 9 more of the same kind; your total is -18 of these apples.

Key Tips for Combining Like Terms

• Look for terms with the same variable and exponent.
• Only add or subtract the coefficients (the numerical part).
• Keep the variable part of the term unchanged.
• Remember that the order in which you combine them does not affect the result due to the commutative property of addition.

Simplify Algebraic Expression

Simplification of an algebraic expression is the process of transforming it into the simplest form possible. The goal is to make the expression easier to read and work with. Take our example \( -9 w^{5} - 9 w^{4} - 9 w^{5} + 10 w^{4} \), after combining like terms, you end up with \( -18w^5 + w^4 \). This expression is simpler because it has fewer terms and is more straightforward to interpret or further manipulate.

The beauty of simplification lies in the fact that although the expression looks different, it still has the same value as the original expression. It's akin to cleaning up a messy room; the room is the same size, but it's just much tidier and more comfortable to use.

Steps for Simplifying an Algebraic Expression

• Combine like terms, as previously discussed.
• Use the distributive property if necessary to eliminate parentheses by multiplying.
• Simplify any complex fractions or expressions within the terms.
• Ensure all constants and known values are added or subtracted to get a final figure.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with solving equations and understanding the rules that govern operations with variables. When working with expressions like \( -9 w^{5} - 9 w^{4} - 9 w^{5} + 10 w^{4} \), the principles of elementary algebra, like combining like terms and simplifying expressions, come into play.

It's about understanding and applying the basic concepts to manipulate algebraic equations and expressions, laying the groundwork for more advanced math topics. Key skills include operations with real numbers, understanding properties of operations (like the distributive property), and the ability to handle variables and constants.

Foundations of Elementary Algebra

• Know the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS).
• Understand the concept of variables and how they can represent numbers.
• Realize that algebraic expressions can be manipulated using various properties, while the equality holds true.
• Develop the ability to translate word problems into algebraic expressions and equations.