Problem 29 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ -5^{2}-4^{2} $$

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Understand the Expression

The given expression is \[-5^2 - 4^2\].We need to calculate each power separately and then subtract them.

Step 2: Calculate the Power of -5

The expression has \(-5^2\),which means \((-5) imes (-5)\).This equals \(25\).

Step 3: Calculate the Power of 4

Next, calculate \(4^2\),which is \(4 imes 4\). This equals \(16\).

Step 4: Perform Subtraction

Now you subtract the results of each power:\[25 - 16 = 9\]

Key Concepts

Numerical Expressions

Numerical expressions are mathematical phrases that involve numbers and operational symbols but do not require an equal sign. They can include addition, subtraction, multiplication, division, and exponents. In the context of the task, the expression \[-5^2 - 4^2\] consists of numerical values, an exponent operation, and a subtraction. Understanding how to work with numerical expressions is essential when simplifying them, as it helps organize and resolve different parts of the problem. To get started on simplifying any numerical expression, follow these steps:

• Identify and separate each component of the expression, such as individual numbers and operators.
• Resolve operations inside parentheses first, if applicable.
• Know the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

This framework will provide clear guidance through the problem-solving process.

Simplification

Simplification is the process of making a mathematical expression as straightforward and concise as possible, without changing its value. For expressions involving multiple operations, simplification often involves evaluating parts methodically according to the order of operations, until the expression cannot be further reduced. In \[-5^2 - 4^2\], we simplify by first addressing the exponent operations for both terms. Simplification isn't about changing what the expression represents but solving it step-by-step:

• Begin by evaluating any exponents in the expression, such as \(-5^2\) and \(4^2\).
• Proceeding with the calculations, this leads to \((-5) \times (-5) = 25\) and \(4 \times 4 = 16\).
• The next simplification step is to perform any remaining operations like addition or subtraction, which in this case means subtracting: \(25 - 16 = 9\).

Through systematic simplification, we arrive at the precise and most simplified answer.

Powers

Powers, or exponents, play a crucial role in simplifying numerical expressions. A power is an expression that denotes repeated multiplication of a number by itself. Understanding powers requires knowing the key terms: the base (the number that is being multiplied) and the exponent (the number of times the base is multiplied by itself). For instance, in \(5^2\), "5" is the base and "2" is the exponent, signifying that 5 is multiplied by itself once: \(5 \times 5\).In our exercise:

• The first power \(-5^2\) is calculated by squaring \(-5\), and while this might seem tricky due to the negative sign, remember that squaring a negative results in a positive: \((-5) \times (-5) = 25\).
• The second power \(4^2\) represents the straightforward task of \(4 \times 4 = 16\).

Understanding how to handle powers and remembering the rules of signs and arithmetic helps simplify complex expressions accurately and efficiently.