Simplifying Expressions Distributive Property And Combining Like Terms

In mathematics, simplifying expressions is a fundamental skill. It involves making an expression as concise and easy to understand as possible without changing its value. One common technique for simplifying algebraic expressions involves using the distributive property to remove parentheses and then combining like terms. This article will guide you through the process, step by step, using the expression: $-6b(7a - 2b) + 7(-5ab - 8b^2)$ as an example.

Understanding the Distributive Property

The distributive property is a powerful tool that allows us to multiply a single term by multiple terms inside parentheses. It states that for any numbers a, b, and c:

$a(b + c) = ab + ac$

In simpler terms, you multiply the term outside the parentheses by each term inside the parentheses. This property is crucial for removing parentheses and simplifying expressions. Let’s consider a basic example:

$3(x + 2)$ can be simplified by distributing the 3 across both terms inside the parentheses:

$3 * x + 3 * 2 = 3x + 6$

This principle extends to more complex expressions, including those with variables and coefficients. The key is to ensure that the term outside the parentheses is multiplied by every term inside the parentheses, paying close attention to signs (positive and negative).

Applying the Distributive Property to Our Expression

Now, let's apply the distributive property to our original expression: $-6b(7a - 2b) + 7(-5ab - 8b^2)$. We have two sets of parentheses, each requiring distribution. First, we distribute $-6b$ across $(7a - 2b)$, and then we distribute $7$ across $(-5ab - 8b^2)$.

1. Distribute $-6b$: $-6b * 7a = -42ab$

$-6b * -2b = 12b^2$

So, $-6b(7a - 2b)$ becomes $-42ab + 12b^2$.

2. Distribute $7$: $7 * -5ab = -35ab$

$7 * -8b^2 = -56b^2$

Thus, $7(-5ab - 8b^2)$ becomes $-35ab - 56b^2$.

Now, our expression looks like this: $-42ab + 12b^2 - 35ab - 56b^2$. The parentheses have been successfully removed, and we’re ready for the next step: combining like terms.

Identifying and Combining Like Terms

Like terms are terms that have the same variables raised to the same powers. For example, $3x^2$ and $-5x^2$ are like terms because they both have the variable $x$ raised to the power of 2. However, $3x^2$ and $3x$ are not like terms because the powers of $x$ are different. Similarly, $2xy$ and $-4xy$ are like terms, while $2xy$ and $2x$ are not.

Combining like terms involves adding or subtracting their coefficients (the numbers in front of the variables) while keeping the variable part the same. For instance, $3x^2 + (-5x^2)$ simplifies to $-2x^2$ because we add the coefficients $3$ and $-5$ to get $-2$, and the variable part $x^2$ remains unchanged.

Combining Like Terms in Our Expression

Looking at our expression, $-42ab + 12b^2 - 35ab - 56b^2$, we can identify two sets of like terms:

1. $ab$ terms: $-42ab$ and $-35ab$

2. $b^2$ terms: $12b^2$ and $-56b^2$

Now, let's combine these like terms:

1. Combining $ab$ terms: $-42ab + (-35ab) = -42ab - 35ab = -77ab$

2. Combining $b^2$ terms: $12b^2 + (-56b^2) = 12b^2 - 56b^2 = -44b^2$

After combining like terms, our expression simplifies to $-77ab - 44b^2$.

Final Simplified Expression

By applying the distributive property and combining like terms, we have successfully simplified the original expression. The simplified form is:

$-77ab - 44b^2$

This expression is now in its simplest form, meaning there are no more like terms to combine and no parentheses to remove. This simplified form is equivalent to the original expression but is much easier to work with in further mathematical operations or analysis.

Step-by-Step Recap

To summarize, simplifying expressions using the distributive property and combining like terms involves the following steps:

1. Apply the Distributive Property: Multiply the term outside the parentheses by each term inside the parentheses. Be careful to consider the signs (positive or negative) of the terms.

2. Identify Like Terms: Look for terms that have the same variables raised to the same powers.

3. Combine Like Terms: Add or subtract the coefficients of the like terms while keeping the variable part the same.

Let’s briefly review the steps we took with our example expression, $-6b(7a - 2b) + 7(-5ab - 8b^2)$:

1. Distribute $-6b$ and $7$:

$-6b(7a - 2b) + 7(-5ab - 8b^2) = -42ab + 12b^2 - 35ab - 56b^2$

2. Identify Like Terms:

Like terms are $-42ab$ and $-35ab$, and $12b^2$ and $-56b^2$.

3. Combine Like Terms:

$-42ab - 35ab = -77ab$

$12b^2 - 56b^2 = -44b^2$

4. Final Simplified Expression:

$-77ab - 44b^2$

By following these steps, you can simplify a wide range of algebraic expressions.

Additional Tips and Considerations

Double-Checking Your Work

It’s always a good practice to double-check your work, especially in mathematics. A common method for verifying your simplification is to substitute numerical values for the variables in both the original and simplified expressions. If the expressions are equivalent, they should yield the same result for any given set of values. For instance, let’s substitute $a = 1$ and $b = 2$ into both the original and simplified expressions:

1. Original Expression: $-6b(7a - 2b) + 7(-5ab - 8b^2) = -6(2)(7(1) - 2(2)) + 7(-5(1)(2) - 8(2)^2)$ $ = -12(7 - 4) + 7(-10 - 32)$$ = -12(3) + 7(-42)$$ = -36 - 294$$ = -330$

2. Simplified Expression: $-77ab - 44b^2 = -77(1)(2) - 44(2)^2$ $ = -154 - 44(4)$$ = -154 - 176$$ = -330$

Since both expressions yield the same result (-330), we can be confident that our simplification is correct. This method provides a practical way to catch errors and ensure accuracy.

Handling More Complex Expressions

The principles we’ve discussed can be applied to more complex expressions involving multiple sets of parentheses, various variables, and higher powers. The key is to systematically apply the distributive property, paying close attention to the order of operations (PEMDAS/BODMAS). When dealing with nested parentheses, it’s often best to start with the innermost set and work your way outwards. For example, consider the expression:

$2[3x - (4y + 2(x - y))]$

To simplify this, you would first distribute the 2 inside the innermost parentheses, then combine like terms within the inner set, and finally, distribute and combine like terms in the outer set of brackets.

Common Mistakes to Avoid

1. Sign Errors: A very common mistake is mishandling negative signs when distributing. Always ensure that the negative sign is correctly applied to each term inside the parentheses.

2. Incorrect Distribution: Another common error is failing to multiply every term inside the parentheses by the term outside. Make sure each term is accounted for.

3. Combining Non-Like Terms: Remember that you can only combine terms that have the same variables raised to the same powers. Avoid the mistake of combining terms like $3x^2$ and $3x$.

4. Forgetting to Simplify Completely: Always ensure that your final expression is fully simplified, meaning there are no more like terms to combine or distributions to perform.

By being mindful of these common pitfalls, you can significantly reduce errors and improve your accuracy in simplifying algebraic expressions.

Conclusion

Simplifying expressions using the distributive property and combining like terms is a foundational skill in algebra. By mastering these techniques, you can make complex expressions more manageable and easier to work with. Remember to apply the distributive property carefully, identify and combine like terms accurately, and double-check your work to avoid common mistakes. With practice, you’ll become proficient in simplifying a wide variety of algebraic expressions, which is essential for success in more advanced mathematical topics.

In summary, the simplified form of the expression $-6b(7a - 2b) + 7(-5ab - 8b^2)$ is $-77ab - 44b^2$. This process highlights the importance of the distributive property and combining like terms in algebraic simplification.