Problem 49 - Elementary Algebra

Question

Simplify. $$ (7+4 *(-2)) /\left(-3+(-2)^{\wedge} 2\right) $$

Step-by-Step Solution

Answer

\(-1\)

Step 1: Simplify Inside Parentheses

First, evaluate any expressions inside the parentheses. In this problem, compute the term within the parentheses in the numerator: \(4 \times (-2)\). This equals \(-8\). Thus, the numerator becomes \(7 + (-8)\).

Step 2: Simplify Numerator

Simplify the expression in the numerator. Perform the addition: \(7 + (-8) = 7 - 8 = -1\). So, the numerator simplifies to \(-1\).

Step 3: Evaluate Exponent in Denominator

Now, compute the exponent in the denominator. Calculating \((-2)^2\) results in \(4\).

Step 4: Simplify Denominator

Simplify the expression in the denominator. Combine terms: \(-3 + 4 = 1\).

Step 5: Division Operation

Now, divide the simplified numerator by the simplified denominator: \(-1 / 1 = -1\).

Step 6: Final Simplified Expression

The expression simplifies completely to \(-1\).

Key Concepts

Parentheses Evaluation

When you see parentheses in a math expression, it's like a signal that says, "solve this part first!" Parentheses help group numbers and operations, so we process them correctly. In our example, the initial task is to evaluate everything within the parentheses of the numerator and the denominator separately.
For the numerator, you have \(7 + 4 imes (-2)\). It's important to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction, left to right). First, multiply \(4\) by \(-2\), giving you \(-8\). Then add \(7\) to \(-8\), simplifying it to \(-1\).
This step ensures that you've clarified what the numbers actually mean before you do anything else. Neglecting this can lead to incorrect simplifications, and your final answer won't be right.

Numerator Simplification

After evaluating what's inside the parentheses, it's crucial to simplify the expression left in the numerator. In our example, once you've done the multiplication inside the parentheses, you'll be left with \(7 + (-8)\).
Now, this is a straightforward arithmetic operation: just add the two numbers. Adding \(7\) and \(-8\) is like subtracting \(8\) from \(7\), which gives you \(-1\).
Breaking it down into steps:

• Recognize that \(7 + (-8)\) is the same as \(7 - 8\).
• Perform the subtraction: \(7 - 8 = -1\).

Thus, you have successfully simplified the numerator to \(-1\). This part is crucial as mistakes in calculations here will follow through to the final result.

Denominator Simplification

Moving to the denominator requires both evaluation of an exponent and simplification. You have to handle \(-3 + (-2)^2\) appropriately. Begin by evaluating the exponent first, because according to the order of operations, exponents are handled before addition or subtraction.
Calculating \((-2)^2\) means multiplying \(-2\) by \(-2\), which results in \(4\). Once this is completed, you address the rest of the expression by adding \(-3 + 4\).
Here’s a quick breakdown:

• Calculate the square of \(-2\), which is \(4\).
• Add the result to \(-3\): \(-3 + 4 = 1\).

Now, your expression in the denominator is simplified to \(1\). Properly handling these steps sets the stage for an accurate division step.

Division Operation

Now that both the numerator and the denominator are simplified, you're ready to perform the division operation. In our case, you have simplified the fraction to \(-1 / 1\).
This division is straightforward:

• When you divide a number by \(1\), the result is the same number.
• In our expression, dividing \(-1\) by \(1\) gives you \(-1\).

Division simplifies the fraction into a single, clean number, wrapping up the simplification process neatly. Always ensure both the numerator and denominator are fully simplified before performing division to avoid errors.

Exponent Evaluation

Understanding and performing exponent evaluation correctly is essential for algebra simplification. In our original expression, the exponent we need to evaluate is \((-2)^2\), which appears in the denominator.
When evaluating an exponent like \((-2)^2\), remember that it means \(-2\) multiplied by itself. Here's the step-by-step breakdown:

• Negative sign squared turns positive: \((-2)\times(-2) = 4\).

This evaluation is vital because it determines how the numbers in the denominator interact with each other, which affects the entire fraction. Mistakes here can lead to incorrect simplifications and, therefore, inaccurate results.