Problem 52 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 4 a^{2} b c^{3}+5 a b c^{3}+9 a b c^{3}+7 a^{2} b c^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(18 a b c^3 + 7 a^{2} b c^{2}\).

Step 1: Identify like terms

Identify terms with the same variable part, which would be \(a b c^3\). We can see the first three terms share this same variable part. $$ 4 a^{2} b c^{3}+5 a b c^{3}+9 a b c^{3}+(7 a^{2} b c^{2}) $$

Step 2: Combine coefficients of like terms

Now, add the coefficients of the like terms and keep the same variable parts (in this case, \(a b c^3\)): $$ (4+5+9) a b c^3+(7 a^{2} b c^{2}) $$

Step 3: Simplify the expression

Add the coefficients of the like terms and simplify the expression: $$ 18 a b c^3+7 a^{2} b c^{2} $$ The simplified algebraic expression is: $$ 18 a b c^3+7 a^{2} b c^{2} $$

Key Concepts

Understanding Like Terms

In algebra, 'like terms' are terms that contain the same variables raised to the same power. For instance, in the expression 4a^2bc^3 + 5abc^3, the terms 5abc^3 and 9abc^3 are like terms because they both have the combination of variables abc with the same exponents. The concept of like terms is crucial because it allows us to simplify algebraic expressions.

In the example from our exercise, 4a^2bc^3, 5abc^3, and 9abc^3 seem to be similar at a glance, but in reality, only 5abc^3 and 9abc^3 are like terms. The difference here is the exponent on a. Because ‘like terms’ must match exactly in both the variable parts and their exponents, 4a^2bc^3 does not qualify as a like term with the others. Recognizing like terms accurately is the first step towards simplifying an algebraic expression effectively.

Combining Coefficients of Like Terms

Once you have identified like terms, the next step is to combine their coefficients. Coefficients are the numerical factors in front of variables. For example, in the like terms 5abc^3 and 9abc^3, the coefficients are 5 and 9 respectively. When combining, we only add or subtract these numerical coefficients, keeping the variable part of the term unchanged.

So for our exercise, the like terms 5abc^3 and 9abc^3 would be combined by adding their coefficients to get (5+9)abc^3, which simplifies to 14abc^3. It is important to remember that only the coefficients are combined; the variable part remains the same. This is because the coefficients represent the quantity of the item being described by the variable part, and when you have like terms, you're essentially adding or subtracting the same items.

The Process of Algebraic Expression Simplification

Algebraic expression simplification involves reducing an expression into its simplest form, making it more comprehensible and easier to work with. This process often involves several operations, including combining like terms, as we've previously discussed. The ultimate goal is to have an expression with as few terms as possible without changing its value.

In step three of our solved example, after identifying and combining like terms, we arrived at 18abc^3 + 7a^2bc^2, which is a simplified version of the original expression. This final expression cannot be simplified any further because there are no like terms to combine. It's essential for students to practice simplification regularly as it is widely used in solving equations, factoring expressions, and working with polynomials, making it a cornerstone of algebra.