Problem 61 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-7n^2 + n + 6\).

Step 1: Simplify the innermost parentheses

Begin by looking at the expression inside the innermost parentheses: \(-4n^2 + n + 6\). There are no further terms to simplify inside these parentheses, so we can consider it as is.

Step 2: Remove the second layer of parentheses

Now look at the next set of parentheses that encloses \(n^2 - (-4n^2 + n + 6)\). Distribute the negative sign through the expression: \(n^2 - (-4n^2 + n + 6) = n^2 + 4n^2 - n - 6\). Simplify by combining like terms: \((n^2 + 4n^2) - n - 6 = 5n^2 - n - 6\).

Step 3: Remove the outermost parentheses

With the expression \(-2n^2 - [5n^2 - n - 6]\), distribute the negative sign over the brackets: \(-2n^2 - 5n^2 + n + 6\). Combine like terms: \((-2n^2 - 5n^2) + n + 6 = -7n^2 + n + 6\).

Step 4: Final simplification

The expression is now \(-7n^2 + n + 6\). There are no like terms left to combine. This is the simplified form.

Key Concepts

Distributive Property

The Distributive Property is an essential algebraic tool used to simplify expressions involving parentheses. It allows you to distribute a factor across terms inside the parentheses and is represented as: \( a(b + c) = ab + ac \).
This property can be used to remove parentheses and simplify expressions.

For the exercise, the Distributive Property is applied twice. First, in Step 2, the expression inside the brackets is \(n^2 - (-4n^2 + n + 6)\). The negative sign outside the parentheses must be distributed to each term inside, effectively reversing their signs:

• \(-(-4n^2) = 4n^2\)
• \(-n = -n\)
• \(-6 = -6\)

The result is \(n^2 + 4n^2 - n - 6\).
Next, in Step 3, the Distributive Property is applied to the outer expression \(-2n^2 - [5n^2 - n - 6]\). Again, distribute the negative sign across all terms to get:

• \(-5n^2 = -5n^2\)
• \(-(-n) = +n\)
• \(-(-6) = +6\)

This gives us \(-2n^2 - 5n^2 + n + 6\).

Combining Like Terms

Combining Like Terms is another crucial technique in algebraic simplification. It helps in gathering similar terms to make an expression simpler and more concise.

Like terms are terms that contain the same variables raised to the same power. For example, \(3x^2\) and \(-5x^2\) are like terms because both include \(x^2\). Terms without variables are also like terms, such as \(6\) and \(-4\).

In our example, once the Distributive Property had been applied, we were left with the expression \(n^2 + 4n^2 - n - 6\). Here, the like terms are \(n^2\) and \(4n^2\). Adding these gives us \(5n^2\).
The expression becomes \(5n^2 - n - 6\). In the next step, combining terms from the expression \(-2n^2 - 5n^2 + n + 6\), the like terms \(-2n^2\) and \(-5n^2\) result in \(-7n^2\).
Combining all gives \(-7n^2 + n + 6\), which is fully simplified.

Expressions with Parentheses

Expressions with Parentheses are common in algebra and serve to clarify which operations should be performed first. Parentheses can dramatically affect the value of an expression because they indicate important precedence rules in mathematical operations.

The primary rule is to simplify expressions within parentheses first. In our exercise, the innermost parentheses are tackled first: \(-4n^2 + n + 6\). This expression doesn't need further simplification immediately, so it remains as is in the first step.
Once no further simplification is required within the innermost parentheses, move on to the next outer layer. This was done for \(n^2 - (-4n^2 + n + 6)\) by distributing the negative sign.
Finally, apply the operations to the expressions in the outermost parentheses to ensure the whole equation is properly simplified. The sequence of operations based on the parentheses structure keeps calculations orderly and reduces errors.