Problem 32 - Algebra A Combined Function

Question

Perform each indicated operation. $$ (-12)^{2}+(-1)(2)-6 $$

Step-by-Step Solution

Answer

The result is 136.

Step 1: Evaluate the Exponentiation

The expression begins with \((-12)^2\). To perform this, multiply \(-12\) by itself: \(-12 \times -12 = 144\). So, \((-12)^2 = 144\).

Step 2: Perform Multiplication

Next, handle the multiplication \((-1)(2)\). Multiply \(-1\) and \(2\): \(-1 \times 2 = -2\).

Step 3: Substitute Results into Expression

Substitute the results from Steps 1 and 2 back into the expression: \(144 + (-2) - 6\).

Step 4: Perform Addition

Now perform the addition of \(144\) and \(-2\): \(144 + (-2) = 142\).

Step 5: Perform Final Subtraction

Finally, subtract \(6\) from \(142\): \(142 - 6 = 136\).

Key Concepts

Exponentiation

Exponentiation is an important mathematical operation where a number is raised to the power of another number. In simple terms, it means multiplying a number by itself a certain number of times. The expression \((-12)^2\) means multiplying \(-12\) by itself. \(-12 \ imes \ -12 = 144\). Keep in mind:

• An even power (such as 2) of a negative number (like -12) results in a positive outcome. This is why the square of \(-12\) is \144\.
• The base number is \(-12\), and the exponent is \2\.

Understanding exponentiation is crucial in various fields of science, engineering, and finance. Recognizing the behavior of negative bases is fundamental to solving problems correctly.
Always square the entire base including the sign, as shown in the problem.

Multiplication

Multiplication is the process of adding a number to itself repeatedly. In our expression, we have \((-1)(2)\), which means \-1\ is multiplied by \2\. To compute:

• The product of \-1\ and \2\ is \-2\, since multiplying any number by \-1\ inverts its sign.
• This shows a pivotal property of multiplication with negative numbers. The result takes the negative sign of \-1\.

Multiplication plays a crucial role in simplifying expressions. It appears in various scenarios from day-to-day calculations to complex algebraic equations.
In problems involving multiple operations, multiplication is usually calculated before addition and subtraction according to the order of operations.

Addition and Subtraction

Addition and subtraction are fundamental arithmetic operations. In the expression \(144 + (-2) - 6\), we need to handle these with care:

• Adding \(-2\) is like subtracting \2\: \144 + (-2) = 142\.
• After applying addition, subtract \6\ from \142\ to get \136\.

When dealing with a series of additions and subtractions, it's important to move left to right following the operations order: first handle addition then subtraction. Handling positive and negative numbers effectively can either result in incrementing or decrementing a value.
This procedure ensures correct computation and is essential for maintaining numerical accuracy.

Integer Operations

Integer operations cover addition, subtraction, multiplication, and exponentiation when dealing with whole numbers. Understanding the behavior of integers, particularly with different signs, is critical:

• Positive integers increase the value when added, while negative integers decrease it.
• Multiplying two negative integers results in a positive product, while a negative and a positive produces a negative outcome.
• Subtraction of integers is essentially the addition of a negative value.

Integer operations form the foundation of more advanced mathematics. Engaging with them helps in not only solving equations but also in real-life scenarios where calculations are needed.
Practicing operations involving integers bolsters proficiency in tackling more complex mathematical challenges.