Problem 78 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 6\left\\{m+5 n[n+3(n-1)]+2 n^{2}\right\\}-4 n^{2}-9 m $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression by applying the distributive property and combining like terms: $$6\left\\{m+5 n[n+3(n-1)]+2 n^{2}\right\\}-4 n^{2}-9 m$$ Answer: The simplified algebraic expression is $$-3m + 38n^2 + 90n(n-1)$$

Step 1: Apply the distributive property to the given expression

First, we apply the distributive property by multiplying the term 6 with each term inside the brackets. The given expression is: $$ 6\left\\{m+5 n[n+3(n-1)]+2 n^{2}\right\\}-4 n^{2}-9 m $$ Applying the distributive property, we get: $$ 6m+6(5n[n+3(n-1)])+6(2 n^{2})-4 n^{2}-9 m $$

Step 2: Further expand the expression

Next, we need to expand the expression to further simplify it. Expand the term \(6(5n[n+3(n-1)])\): $$ 6(5n[n+3(n-1)])=30n^2+90n(n-1) $$ Now our expression becomes: $$ 6m+30n^2+90n(n-1)+12n^2-4n^2-9m $$

Step 3: Combine like terms

Now, we will combine the like terms in the expression. We have: 1. \(6m\) and \(-9m\) 2. \(30n^2\), \(12n^2\), and \(-4n^2\) 3. \(90n(n-1)\) Let's combine them: 1. For \(m\) terms: \(6m-9m=-3m\) 2. For \(n^2\) terms: \(30n^2+12n^2-4n^2=38n^2\) 3. For \(n(n-1)\) terms: \(90n(n-1)\) remains as it is Now, our expression becomes: $$ -3m+38n^2+90n(n-1) $$ The simplified algebraic expression is: $$ -3m + 38n^2 + 90n(n-1) $$

Key Concepts

Distributive Property

In the world of algebra, the distributive property is a powerful tool that helps simplify expressions. It's all about spreading multiplication over addition or subtraction within brackets. When you see a term outside a bracket, it needs to be multiplied by everything inside. For example, if you have a term like \(a(b+c)\), you will multiply \(a\) with both \(b\) and \(c\), leading to \(ab + ac\).

In the original problem, the expression that we need to simplify is:

• \(6\{m+5n[n+3(n-1)]+2n^2\}-4n^2-9m\)

To use the distributive property here, we multiply the 6 by each term in the brackets. This is carefully done to ensure each piece inside the bracket receives its fair share of the 6. It's important to track each term separately to avoid mistakes. So, we end up with:

• \(6m+6(5n[n+3(n-1)])+6(2n^2)\)

This step sets the stage for easier simplification and expansion, building a strong foundation for further algebraic processes.

Combining Like Terms

After distributing terms and expanding expressions, the next key concept is combining like terms. This process bundles up terms that have identical variables and powers, simplifying the equation. If terms share the same variable raised to the same power, you can mathematically add or subtract them just like regular numbers.

In our simplified expression, we identified terms that can be combined:

• \(6m\) and \(-9m\)
• \(30n^2\), \(12n^2\), and \(-4n^2\)

For the terms with \(m\), we have \(6m - 9m = -3m\). Similarly, for the \(n^2\) terms, combining \(30n^2 + 12n^2 - 4n^2\) yields \(38n^2\). Keeping terms organized helps eliminate clutter and reveals a simpler expression. Notice how \(90n(n-1)\) was untouched, as it didn’t have similar terms to combine with. These steps bring the expression closer to its simplest form, bridging toward an easy-to-read result.

Algebraic Expansion

Algebraic expansion is the method of removing parentheses by multiplying out the terms inside, which helps break down more complex expressions into manageable parts. This technique unfolds expressions like a book, making what's inside visible and clearer.

In the original exercise, implementing algebraic expansion was necessary to simplify terms such as \(6(5n[n+3(n-1)])\). When expanded, it revealed \(30n^2 + 90n(n-1)\), turning intricate nested terms into simpler expressions. This unfolding acts as a bridge, allowing you to connect distributed and combined like terms comfortably.

By expanding the expression, we smoothly transition into an easier path for simplification and combining like terms. Each step uncovers a clearer path to the final, comprehensive form. As algebraic expressions grow more complex, mastering this process equips you with the confidence and skill to tackle a wide range of algebraic equations.