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Chapter 2: Problem 34

For Exercises \(1-74,\) simplify. $$2-(-5)(-4)^{2}$$

Short Answer

Expert verified
Answer: The simplified form of the expression is $$-78$$.

Step by step solution

01

Evaluate the Exponent Inside the Parentheses

Start by solving for the exponent inside the parentheses. To do this, raise \(-4\) to the power of \(2\). $$(-4)^{2}=16$$
02

Perform the Multiplication

Now that we have the result of the exponent, we can perform the multiplication: $$-(-5)(16)$$ Since two negatives make a positive, the multiplication becomes: $$5(16) = 80$$
03

Perform the Subtraction

Now that we have the result of the multiplication, we can perform the subtraction: $$2 - 80 = -78$$ So, the simplified expression is: $$2-(-5)(-4)^{2} = -78$$

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Key Concepts

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

Order of Operations
Understanding the order of operations is essential for solving algebraic expressions. The order of operations is a set of rules that defines the correct sequence to perform operations in an expression. This order ensures accuracy and consistency in mathematical calculations. A helpful acronym to remember the order is **PEMDAS**:
  • **P**arentheses - First, solve expressions inside parentheses.
  • **E**xponents - Next, evaluate exponents (powers).
  • **M**ultiplication and **D**ivision - Perform these operations from left to right.
  • **A**ddition and **S**ubtraction - Finally, complete these operations from left to right.
Applying these rules, we first addressed the exponent \((-4)^2\) in the problem.
Exponents
When working with exponents in algebra, they indicate how many times a number, known as the base, is multiplied by itself. For example, in \((-4)^2\), -4 is the base and 2 is the exponent. Raising a negative base to an even power (like 2 in this case) results in a positive number. So, \((-4)^2 = 16\). Understanding this property of exponents ensures correct simplification of expressions, especially when dealing with negative bases.
Negative Numbers
Dealing with negative numbers can initially seem confusing, but some straightforward rules help simplify such expressions. A key rule is that the product of two negative numbers is positive. This rule must be applied correctly in operations. In the expression \(-(-5) \times 16\), we encountered this rule:
  • The double negative is converted to positive, resulting in \(5 \times 16\).
Recognizing when and how negatives interact with other numbers is critical to avoiding errors.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and mathematical operations. Simplifying these expressions involves following the order of operations and applying properties of numbers, such as rules for exponents or negative numbers. The given expression \(2 -(-5)(-4)^2\) is such an example. To simplify:
  • Evaluate exponents first: \((-4)^2 = 16\).
  • Convert double negatives and perform multiplication: \(5 \times 16 = 80\).
  • Complete with subtraction: \(2 - 80 = -78\).
When simplifying algebraic expressions, the process involves attentively following each step and rule to reach the correct solution.

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Explain the difference between \(-(-3)^{2}\) and \([-(3)]^{2}\)
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