Problem 44 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$(4-9)^{2}$$

Step-by-Step Solution

Answer

25

Step 1: Subtract the Numbers Inside the Parentheses

Calculate the difference between 4 and 9 inside the parentheses. \[ 4 - 9 = -5 \]

Step 2: Square the Result from Step 1

Now, take the result from Step 1, which is \(-5\), and square it.\[ (-5)^{2} = 25 \]

Step 3: Conclusion of Simplification

Thus, the simplified form of \((4-9)^2\) is \(25\).

Key Concepts

Order of Operations

The order of operations is a fundamental concept in mathematics that dictates the sequence in which we solve different parts of a mathematical expression. It is crucial to follow these rules to ensure accurate results. The order is often remembered using the acronym PEMDAS, which stands for:

• Parentheses
• Exponents (or powers, roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our example of simplifying \[(4-9)^{2}\]we begin with any operations inside the parentheses first. This step is crucial as it lays the groundwork for the correct order in further steps. Following the order of operations ensures that we handle each component of an expression logically and accurately. It prevents errors that can arise from performing operations out of sequence.

Working With Parentheses

Parentheses are used in expressions to indicate which operations should be performed first. They act like a spotlight, showing us the priority area within the math problem. When you see parentheses, you solve these parts of the expression first before moving on to other operations. In our exercise, the expression is:

• Inside the parentheses: \(4 - 9\)

By performing the subtraction inside the parentheses, \[4 - 9 = -5,\]we hone in on the essential operation that determines the next step. Parentheses help in guiding our calculations to achieve the correct order, improving our accuracy and understanding. Always check for parentheses in an expression and solve these components first.

Exponentiation

Exponentiation refers to the operation of raising a number, known as the base, to a certain power, the exponent. This means repeated multiplication of the base. For the problem at hand:

• After simplifying inside the parentheses, we have: \(-5\).
• We then raise \(-5\) to the power of 2, written as \((-5)^2\).

The calculation \[(-5)^2 = -5 \times -5 = 25\]demonstrates how exponentiation is applied. Squaring a negative number results in a positive product because multiplying two negative numbers yields a positive result. Hence, exponentiation helps us manage these repeated multiplication tasks simply, revealing the power of concise, compact notation in math. It’s essential to recognize that the position and sign of numbers within parentheses influence the final result of exponentiation.