Simplifying Expressions How To Multiply 6(-5t - 8)

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In the realm of algebra, simplifying expressions is a fundamental skill. This process often involves applying the distributive property, a powerful tool that allows us to efficiently multiply a single term by a group of terms enclosed in parentheses. In this article, we will delve into the step-by-step process of simplifying the expression 6(-5t - 8), ensuring a clear understanding of the underlying principles and techniques.

Understanding the Distributive Property

The distributive property is a cornerstone of algebraic manipulation, enabling us to expand expressions and combine like terms. At its core, the distributive property states that multiplying a number by a sum or difference is the same as multiplying the number by each term within the parentheses individually, and then adding or subtracting the results.

Mathematically, this can be represented as:

  • a(b + c) = ab + ac
  • a(b - c) = ab - ac

Where 'a' is the term outside the parentheses, and 'b' and 'c' are the terms inside the parentheses. The distributive property allows us to eliminate parentheses and simplify expressions, making them easier to work with.

Applying the Distributive Property to 6(-5t - 8)

Let's apply the distributive property to the expression 6(-5t - 8). Here, 6 is the term outside the parentheses, and (-5t - 8) is the expression inside the parentheses. Following the distributive property, we multiply 6 by each term inside the parentheses:

  1. Multiply 6 by -5t: 6 * (-5t) = -30t
  2. Multiply 6 by -8: 6 * (-8) = -48

Now, we combine the results:

-30t + (-48)

Simplifying further, we get:

-30t - 48

Therefore, the simplified form of the expression 6(-5t - 8) is -30t - 48. This process demonstrates how the distributive property allows us to break down complex expressions into simpler, more manageable forms.

Step-by-Step Breakdown

To solidify your understanding, let's break down the simplification process into a step-by-step guide:

Step 1: Identify the term outside the parentheses and the terms inside the parentheses.

In the expression 6(-5t - 8), the term outside the parentheses is 6, and the terms inside the parentheses are -5t and -8.

Step 2: Apply the distributive property by multiplying the term outside the parentheses by each term inside the parentheses.

  • Multiply 6 by -5t: 6 * (-5t) = -30t
  • Multiply 6 by -8: 6 * (-8) = -48

Step 3: Combine the results.

-30t + (-48)

Step 4: Simplify the expression by removing unnecessary parentheses or combining like terms.

-30t - 48

By following these steps, you can confidently simplify expressions using the distributive property.

Common Mistakes to Avoid

While the distributive property is relatively straightforward, there are some common mistakes to watch out for:

  1. Forgetting to distribute to all terms: Ensure that you multiply the term outside the parentheses by every term inside the parentheses. For example, in the expression 6(-5t - 8), you must multiply 6 by both -5t and -8.
  2. Incorrectly applying signs: Pay close attention to the signs (positive or negative) of the terms. A negative number multiplied by a negative number results in a positive number, while a positive number multiplied by a negative number results in a negative number.
  3. Combining unlike terms: Only combine terms that have the same variable and exponent. For example, you cannot combine -30t and -48 because they have different variables (t and a constant).

By being mindful of these common mistakes, you can improve your accuracy and avoid errors when simplifying expressions.

Examples and Practice Problems

To further enhance your understanding, let's explore some additional examples and practice problems.

Example 1: Simplify the expression -2(3x + 5)

  1. Multiply -2 by 3x: -2 * (3x) = -6x
  2. Multiply -2 by 5: -2 * 5 = -10
  3. Combine the results: -6x - 10

Therefore, the simplified form of the expression -2(3x + 5) is -6x - 10.

Example 2: Simplify the expression 4(2y - 7)

  1. Multiply 4 by 2y: 4 * (2y) = 8y
  2. Multiply 4 by -7: 4 * (-7) = -28
  3. Combine the results: 8y - 28

Therefore, the simplified form of the expression 4(2y - 7) is 8y - 28.

Practice Problems:

  1. Simplify the expression 5(4a + 9)
  2. Simplify the expression -3(2b - 6)
  3. Simplify the expression 7(-x + 3)

By working through these examples and practice problems, you can solidify your understanding of the distributive property and improve your ability to simplify expressions.

Real-World Applications

The distributive property is not just a theoretical concept; it has practical applications in various real-world scenarios. Here are a few examples:

  1. Calculating the cost of multiple items: Suppose you want to buy 3 shirts that cost $15 each and 2 pairs of pants that cost $30 each. You can use the distributive property to calculate the total cost: 3($15) + 2($30) = $45 + $60 = $105. This can also be represented as an expression 3(15 + 2(30)), then simplified using the distributive property.
  2. Determining the area of a composite shape: If you have a rectangular garden with a length of (x + 5) meters and a width of 4 meters, you can use the distributive property to calculate the area: 4(x + 5) = 4x + 20 square meters.
  3. Budgeting and financial planning: When planning a budget, you might need to calculate the total cost of various expenses. The distributive property can help you simplify these calculations and make informed financial decisions.

These examples demonstrate how the distributive property can be applied to solve practical problems in everyday life.

Conclusion

Mastering the distributive property is crucial for simplifying algebraic expressions and solving mathematical problems. By understanding the principles behind the distributive property and practicing its application, you can confidently tackle a wide range of algebraic challenges. In this article, we have explored the step-by-step process of simplifying the expression 6(-5t - 8), highlighting the importance of distributing to all terms, paying attention to signs, and combining like terms correctly. With consistent practice and a solid understanding of the distributive property, you can unlock your algebraic potential and excel in your mathematical endeavors. Remember, the key to success lies in understanding the fundamental concepts and applying them consistently. So, continue practicing, explore more complex expressions, and embrace the power of the distributive property in your mathematical journey.