Problem 47 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 2 a^{3} b^{2} c+3 a^{2} b^{2} c^{0}+4 a^{2} b^{2}-a^{3} b^{2} c, \quad c \neq 0 $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$ 2 a^{3} b^{2} c+3 a^{2} b^{2} c^{0}+4 a^{2} b^{2}-a^{3} b^{2} c $$ Answer: $$ a^{3} b^{2} c + 7 a^{2} b^{2} $$

Step 1: Replace \(c^0\) with 1

Since any non-zero variable raised to the power of 0 is equal to 1, we can replace \(c^0\) with 1 in the expression: $$ 2 a^{3} b^{2} c+3 a^{2} b^{2} (1)+4 a^{2} b^{2}-a^{3} b^{2} c $$

Step 2: Simplify the expression

Now, we can simplify the expression by combining the like terms, that is, the terms with the same variables and exponents. Those terms are \(2 a^{3} b^{2} c\) and \(-a^{3} b^{2} c\): $$ (2 a^{3} b^{2} c - a^{3} b^{2} c) + 3 a^{2} b^{2} + 4 a^{2} b^{2} $$

Step 3: Combine the like terms

We can now combine the like terms by taking their sum: $$ a^{3} b^{2} c + 3 a^{2} b^{2} + 4 a^{2} b^{2} $$

Step 4: Factor out the common factors

Now, we can factor out the common factors in the terms. The common factor for \(3 a^{2} b^{2}\) and \(4 a^{2} b^{2}\) is \(a^{2} b^{2}\), so we factor it out: $$ a^{3} b^{2} c + a^{2} b^{2} (3 + 4) $$

Step 5: Simplify further

Finally, we can simplify the expression inside the parentheses: $$ a^{3} b^{2} c + a^{2} b^{2} (7) $$ So, the simplified algebraic expression is: $$ a^{3} b^{2} c + 7 a^{2} b^{2} $$

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the most fundamental strategies is to combine like terms. Like terms are terms that have exactly the same variables raised to the same powers. For instance, in an expression like

\(3x^2y + 5x^2y\), the two terms are considered like terms because they both contain the variable \(x\) squared and \(y\) to the first power.

To combine them, simply add or subtract their coefficients (the numerical part in front of the variables). So \(3x^2y + 5x^2y\) would become \(8x^2y\). This process simplifies the expression, making it easier to work with and solve if it's part of an equation. It's like gathering similar items together when organizing a space—it makes it clearer and more efficient to deal with.

Exponents

Exponents play a crucial role in algebra. They indicate how many times a number, known as the base, is multiplied by itself. For instance, \(a^3\) refers to \(a \times a \times a\).

Understanding how to work with exponents is essential when simplifying expressions. A key rule to remember is that any number, except zero, raised to the power of zero is always one, e.g., \(c^0 = 1\). This rule helps simplify terms where variables could be in the exponent, as seen in the provided exercise. Also, when multiplying like bases, you add the exponents, and when dividing like bases, you subtract the exponents. These rules enable us to manipulate and simplify expressions with exponents effectively.

Factoring in Algebra

Factoring in algebra involves breaking down an expression into simpler, 'factor' components that, when multiplied together, give back the original expression. It's analogous to finding the ingredients in a cooked dish. The aim is to reveal a more straightforward structure of the expression, which can be especially useful for solving equations.

For example, if you have \(4a^2b^2 + 8a^3b^3\), you can factor out \(4a^2b^2\), the greatest common factor, to get \(4a^2b^2(1 + 2ab)\). Factoring can assist in further simplification especially when working with polynomials and is a major step in solving quadratic equations as well.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form. It involves a series of steps that can include distributing, combining like terms, factoring, and canceling. The aim is to make the expression as concise as possible without changing its value.

It's like creating a shorter, clearer path to the same destination. The simplification might involve arithmetic operations or applying laws of exponents, depending on the given expression. Keeping expressions simple is not just aesthetically pleasing—it makes it easier to see relationships between variables and can be crucial when solving for unknowns or in higher-level calculus.