Problem 50 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$4 \pi-(-12 \pi)$$

Step-by-Step Solution

Answer

The result of the operation \(4 \pi-(-12 \pi)\) is \(16 \pi\).

Step 1: Understanding the double negative

In the exercise, have \(4 \pi - (-12 \pi)\). A double negative becomes a positive, so we rewrite the subtraction as addition: \(4 \pi + 12 \pi\)

Step 2: Combining like terms

Now that the operation has been simplified to addition, recognize these as like terms (terms with the same variable to the same power) and combine \(4 \pi\) and \(12 \pi\). This is done by adding the coefficients (the numbers before the \( \pi \) symbol), which gives us \(16 \pi\)

Key Concepts

Combining Like Terms

Understanding how to combine like terms is fundamental in algebra. It simplifies expressions and makes it easier to solve equations. Like terms are terms that have the exact same variables raised to the same power. In the exercise, we have terms involving the mathematical constant \( \( \pi \)\). Here, \(4 \( \pi \)\) and \(12 \( \pi \)\) are like terms because they both have the \( \pi \) symbol. When combining these terms, we only add the coefficients—the numerical part of the terms—while \( \pi \) remains unchanged.

To visualize this, imagine you have 4 apples and someone gives you 12 more apples. The type of fruit doesn’t change; you simply have more of it. Similarly, adding \(4 \( \pi \)\) to \(12 \( \pi \)\) gives us \(16 \( \pi \)\), just as having 4 plus 12 apples gives you 16 apples. Remember that the variables work just like placeholders for something common in both terms, in this case, \( \pi \) is our 'apple'.

Operating with Pi

When working with the constant \( \pi \) in algebra, you treat it like any other variable—except that it represents a special number approximately equal to 3.14159. Because \( \pi \) is irrational (its exact value cannot be represented as a simple fraction), it is often left in its symbol form when we perform mathematical operations.

In our exercise, you'll notice that the subtraction involves only \( \pi \) terms. It is essential to realize that any operation with \( \pi \) follows the ordinary rules of arithmetic: you can add, subtract, multiply, and divide coefficients while leaving \( \pi \) intact. Using \( \pi \) this way actually helps to maintain precision in computations until you need to approximate an actual number, such as when calculating the circumference or area of a circle.

Double Negative in Algebra

A double negative in algebra often causes confusion, but it's quite simple once you understand the rule. In the expression \(4 \( \pi \) - (-12 \( \pi \) )\), the double negative \(--12\) becomes positive \(+12\). Mathematically, subtracting a negative is the same as adding a positive. This rule is a fundamental concept in algebra and gives us a new, equivalent expression: \(4 \( \pi \) + 12 \( \pi \)\).

Think of negation as 'owing' something. If you owe an owed amount, you're effectively canceling both debts out and gaining that amount instead. So, when encountering a double negative, flip the second negative sign to a plus, and proceed with the operation as you normally would with positive numbers.