Problem 60 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$4 x^{2}-\left[-x^{2}-\left(5 x^{2}-6\right)\right]$$

Step-by-Step Solution

Answer

The simplified expression is \(10x^2 - 6\).

Step 1: Analyze the Expression

This expression is \( 4x^2 - \left[ -x^2 - (5x^2 - 6) \right] \). Notice there are nested parentheses, and we'll start by simplifying the innermost expression first.

Step 2: Simplify the Innermost Parentheses

Look at the expression inside the innermost parentheses: \((5x^2 - 6)\). This cannot be simplified further on its own, so we'll later use its value as it is.

Step 3: Remove the Inner Parentheses

Now address \( -x^2 - (5x^2 - 6) \). Distribute the negative sign across the inner parentheses: \[-x^2 - 5x^2 + 6 = -6x^2 + 6 \].

Step 4: Remove the Remaining Parentheses

Now substitute the result of the inner expression back into the remaining expression: \[ 4x^2 - (-6x^2 + 6) \].Again, distribute the negative sign across the terms:\[ 4x^2 + 6x^2 - 6 \].

Step 5: Combine Like Terms

Finally, combine the like terms (terms involving \(x^2\)): \[ 4x^2 + 6x^2 - 6 = 10x^2 - 6 \].

Key Concepts

Nested Parentheses

Nested parentheses in mathematical expressions can sometimes make these equations look more complicated than they really are. The key is to work from the inside out, simplifying step by step. Starting with the innermost pair, you treat each enclosing element like layers of an onion.
Imagine the expression as boxes within boxes; you open the smallest box first. In the given exercise, this smallest box is the expression \((5x^2 - 6)\). There's no further operation needed here, so you use it as is in subsequent simplifications. Moving outward, consider the equation within the brackets more as repeating this simplification structure.

• Identify the innermost parentheses
• Consider how each affects the next inner box (or term)

This methodical approach of unwinding each layer helps to organize and simplify complex expressions like peeling away disruptions one at a time.

Distributive Property

The distributive property is a powerful algebraic tool that lets you simplify expressions by distributing multipliers across terms within parentheses. Applied here, it's used to eliminate the parentheses and make calculations more straightforward. In the step \(-x^2 - (5x^2 - 6)\), the distributive property is applied by distributing the negative sign across the terms inside the parentheses.
This step is crucial because it transforms nested, hard-to-decipher expressions into simpler, more manageable ones. You convert \((5x^2 - 6)\) into \(-5x^2 + 6\) by distributing \(-1\). The same concept applies when you cycle through each nested level.
Ultimately, this property helps to organize and combine pieces of the expression into a single form. In this particular exercise, mastering this operation was critical to simplifying the problem.

Combining Like Terms

Combining like terms is the final step in bringing clarity to an algebraic expression. When you combine terms, you only bring together those that share the exact same variable and exponent. In the problem \(4x^2 + 6x^2 - 6\), combining the \(x^2\) terms leads to simplifying the expression further.
It's important to recognize and categorize terms that can be combined, something like sorting apples with apples and oranges with oranges in a basket. This promotes neatness and prepares the expression for any further calculations or interpretations you might need.

• Identify all terms with the same variable and exponent
• Add or subtract the coefficients

Once combined, these terms reduce to form a cleaner, more concise expression \(10x^2 - 6\). This neat simplification is often the goal in algebra, providing a final, simplified result.