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Chapter 1: Problem 88

Simplify each numerical expression. $$ 5(-1.6)-3(2.7)+5(6.6) $$

Short Answer

Expert verified
The expression simplifies to 16.9.

Step by step solution

01

Distribute the Terms

Distribute the numbers outside the parentheses to the numbers inside them. This means you'll multiply each number:\[5(-1.6) - 3(2.7) + 5(6.6)\]\[= -8 - 8.1 + 33\]
02

Simplify the Expression

Now, simplify the expression by performing the addition and subtraction from left to right:\[-8 - 8.1 + 33\]First, combine \(-8\) and \(-8.1\):\[-8 - 8.1 = -16.1\]Then, add \(-16.1\) and \(33\):\[-16.1 + 33 = 16.9\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental rule in mathematics that allows you to distribute multiplication over addition or subtraction inside parentheses. This means, for an expression like \( a(b + c) \), you can multiply the term outside the parentheses by each term inside the parentheses separately. It becomes \( ab + ac \).
For numerical expressions, this principle helps simplify calculations by breaking them into smaller, more manageable parts. In the given exercise:
  • We have \( 5(-1.6) \), which means 5 is distributed to \(-1.6\).
  • We also have \(-3(2.7)\), where \(-3\) is distributed to 2.7.
  • Lastly, \( 5(6.6) \), where 5 is distributed to 6.6.
This results in: \[5(-1.6) - 3(2.7) + 5(6.6) = -8 - 8.1 + 33.\]
By applying the distributive property, you can tackle complex expressions with ease.
Simplification
Simplification involves reducing an expression to its simplest form. After applying the distributive property, you often end up with a series of arithmetic operations 鈥� addition and subtraction in particular. The goal is to combine these into a single, simplified number.
In our example, after distribution, we have \(-8 - 8.1 + 33\).
Simplification proceeds by:
  • First, performing subtraction: \(-8 - 8.1 = -16.1\)
  • Then, adding this result to 33: \(-16.1 + 33 = 16.9\)
This step-by-step approach ensures every part of the expression is simplified, offering a clear path to the final result. Simplification not only makes expressions easier to work with but also clarifies the actual quantity they represent.
Algebraic Operations
Algebraic operations encompass various mathematical procedures, such as addition, subtraction, multiplication, and division. In numerical expressions, these operations are executed sequentially to arrive at a solution.
In the provided exercise, the key operations include multiplication followed by addition and subtraction.
Here is the detailed breakdown:
  • Multiplication: Each number outside the parentheses multiplies what鈥檚 inside, facilitated by the distributive property. For instance, \( 5(-1.6) = -8 \).
  • Subtraction: When numbers are negative or subtractive, you effectively adjust the total downwards, as seen in \(-8 - 8.1 = -16.1\).
  • Addition: Final numbers are summed up to consolidate the result. In this problem, \(-16.1\) is added to \(33\) to yield \(16.9\).
Understanding algebraic operations is crucial because it lets you know what steps to take and in what order to achieve correct and reliable results.

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