Problem 39 - Elementary Algebra

Question

Rewrite the problem in a simpler form. $$ -[-(-15)] $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$-[-(-15)]$$. Answer: The simplified expression is $$- 15$$.

Step 1: Identify the signs in the expression

In the given expression, we have: $$ -[-(-15)] $$ There are three negative signs: two are outside of the parentheses and one is inside.

Step 2: Deal with the inner parenthesis first

According to the rules of arithmetic, we need to deal with the expressions inside the parentheses first. In this case, we have a negative inside the parenthesis: $$ -(-15) $$ As we know, the product of two negatives is positive. So, the expression inside the parenthesis simplifies to: $$ 15 $$

Step 3: Deal with the outer negative sign

Now that we've simplified the expression inside the parenthesis, we have: $$ -[(+15)] $$ As there is only one negative sign left, we don't need to do any new calculations. The expression simplifies to: $$ - 15 $$

Step 4: Write down the final simplified expression

Now we have simplified $$-[-(-15)]$$. The final expression is: $$ - 15 $$

Key Concepts

Order of Operations

When simplifying expressions, especially those involving multiple operations and signs, it's essential to follow the order of operations. This sequence ensures that everyone solves mathematical problems in a consistent way.

The order of operations can be remembered by the acronym PEMDAS:

• Parentheses first
• Exponents (ie Powers and Square Roots, etc.)
• Multiplication and Division (left-to-right)
• Addition and Subtraction (left-to-right)

In our example with -[-(-15)], we start by addressing the innermost parentheses, -(-15). By respecting the order of operations, we avoid mistakes and simplify expressions correctly. Once inner parentheses are resolved, we then move outward, simplifying subsequent layers of parentheses or other operations, until we're left with the final simplified expression.

Negative Numbers Arithmetic

Arithmetic with negative numbers can often be tricky, but remembering a few key rules can make it easier to work with them. For instance, the product of two negative numbers is positive, which comes in handy in our current problem. Similarly, the product of a positive number and a negative number is negative. When you have a negative sign outside parentheses, it affects whatever is inside the parentheses. So, if there's a single negative number inside, it becomes positive once 'negated' by the outer negative sign.

In the problem -[-(-15)], applying these rules, we notice that two negatives will make a positive, turning -(-15) into +15. Then, the remaining outer negative sign is effectively multiplying by -1, transforming (+15) into -15. Understanding how to handle negative signs in arithmetic allows you to simplify expressions with confidence.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. Simplifying these expressions involves combining like terms and applying arithmetic rules, but without an equals sign, unlike algebraic equations. It's about rewriting the expression in a simpler or more useful form.

In the expression -[-(-15)], there are no variables, but the simplification process mirrors that used in algebra. It requires careful observation and the application of arithmetic rules to simplify the signs and finally get to the simplest form of the number. By practicing simplification of algebraic expressions with various operations and terms, students develop an intuition for steps needed and what a simplified form looks like, which is crucial when working with more complex expressions that do include variables.