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Chapter 3: Problem 46

Rewrite the problem in a simpler form. $$ -[-(17)] $$

Short Answer

Expert verified
Answer: The simplified form of the expression is $$17$$.

Step by step solution

01

Identify the double negative

Look at the expression $$-[-(17)]$$. We can see that there are two consecutive negative signs. One is outside the brackets, and the other inside the brackets.
02

Simplify the double negative

When we have two consecutive negative signs (-), it is the same as multiplying by -1 two times. So, this expression can actually be written as $$(-1) \cdot (-1) \cdot (17)$$. Since \((-1) \cdot (-1) = 1\), this simplifies to $$1 \cdot (17)$$.
03

Rewrite the expression in a simpler form

In the last step, we found the expression to be equal to $$1 \cdot (17)$$. This can now be further simplified, as multiplying by 1 has no effect on the number. So the final answer is $$17$$. The simplified form of the given expression is $$17$$.

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Key Concepts

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

Negative Numbers
Understanding negative numbers is crucial in algebra. Think of them as points on a number line that are to the left of zero, the opposite of positive numbers. When you multiply two negative numbers, they cancel out the negativity and give you a positive result. This is why \( -1 \times -1 = 1 \). This rule helps you to simplify expressions with multiple negative numbers such as \( -[-(a)] \) where \( a \) is any number. A double negative turns the value positive. So, when simplifying expressions with double negatives, remember that two 'wrongs' actually make a 'right' in algebra.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. The expression \( -[-(17)] \) is a simple algebraic expression involving only numbers. With more complex expressions that include variables, it's vital to understand not just what operations to perform, but also the structure of the expression, which includes understanding the role of parentheses. In simplifying algebraic expressions, remove parentheses systematically and pay attention to the signs of each term to avoid errors.
Order of Operations
The order of operations, often remembered by the acronym PEMDAS, tells us the sequence in which to solve parts of a mathematical problem: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following this order is key to accurately simplifying algebraic expressions. In our example, we deal with parentheses first. This ensures that the expression inside the parentheses is addressed before the negative sign outside is considered.
Simplification in Algebra
Simplification in algebra often involves combining like terms, reducing expressions to their simplest form, and eliminating double negatives. By applying basic arithmetic rules and the order of operations, simplification makes equations and expressions easier to work with. In the given exercise, \( -[-(17)] \), we are simplifying a double negative. The simplification process is straightforward here, but remember that as expressions get more complex, each simplification step should be performed with care to ensure accuracy. Always aim for the most concise form possible without altering the value of the original expression.

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