Problem 88 - Introductory Algebra for College Students

Question

In Exercises \(81-88,\) simplify each algebraic expression by removing parentheses and brackets. $$4\left(6 x^{2}-3\right)-\left[2\left(5 x^{2}-1\right)+1\right]$$

Step-by-Step Solution

Answer

The simplified algebraic expression is \(14x^{2} - 11\).

Step 1: Remove Inner Parentheses

Start by multiplying the terms within the inner parentheses:- For the left side of the equation within first set of parentheses, distribute 4 through \(6x^{2}\) and -3. This yields \(24x^{2} - 12\).- For the right side of the equation within the first square bracket, distribute 2 through \(5x^{2}\) and -1, before adding 1. This yields \(10x^{2} - 2 + 1\).

Step 2: Simplify Each Side

Simplify each side separately:- The left side, \(24x^{2} - 12\), stays the same.- The right side simplifies to \(10x^{2} - 1\).

Step 3: Subtract The Expressions

Now, subtract \(10x^{2} - 1\) from \(24x^{2}-12\), treating each term separately:- \(24x^{2} - 10x^{2}\) yields \(14x^{2}\).- \(-12 - (-1)\) simplifies as \(-12 + 1\), which equals to -11.

Step 4: Write the Final Simplified Expression

After completion of above steps, the simplified expression will be \(14x^{2} - 11\).

Key Concepts

Distributive Property

When simplifying algebraic expressions, the distributive property is a fundamental tool that enables us to multiply a single term by each term within a set of parentheses or brackets. It's represented mathematically as \(a(b+c) = ab + ac\). Let's break this down with an example from our exercise.

In the given expression, we have two instances where we apply the distributive property. Firstly, the number 4 is distributed across \(6x^2\) and -3: \[4(6x^{2}-3) = 24x^{2} - 12\]. Similarly, the number 2 is distributed across \(5x^2\) and -1, and then we add 1 to complete the operation: \[2(5x^{2}-1)+1 = 10x^{2} - 2 + 1\]. Understanding how to distribute correctly is crucial for simplifying complex expressions and making them more manageable.

Combining Like Terms

Once you've distributed your terms, the next step is to combine like terms to further simplify the expression. Like terms are terms that have the same variables raised to the same power. In our exercise, after distribution, we combine like terms on each side of the subtraction separately before combining the entire expression.

On the left side, \(24x^{2} - 12\) does not have like terms, so it remains unchanged. However, on the right side, after distribution, \(10x^{2} - 2 + 1\) does have like terms - the constants -2 and +1. Combining these gives us \(10x^{2} - 1\). Recognizing and combining like terms effectively streamlines the process of simplifying algebraic expressions.

Algebraic Expression Subtraction

Subtracting algebraic expressions is the final step in our example. This involves subtracting each corresponding term of one expression from the other. When subtracting expressions, it is important to distribute the negative sign, if any, across all terms of the expression after the subtraction sign.

In our problem, we subtract \(10x^{2} - 1\) from \(24x^{2} - 12\). We do this by subtracting the like terms: \[24x^{2} - 10x^{2} = 14x^{2}\] for the \(x^2\) terms and \[-12 - (-1) = -12 + 1 = -11\] for the constants. Therefore, the final simplified expression is \(14x^{2} - 11\). This step might involve rearranging the terms in order to line up like terms, making it easier to see what should be subtracted from what, emphasizing the importance of both organization and attention to detail when simplifying algebraic expressions through subtraction.