Problem 117 - College Algebra

Question

Use the order of operations to simplify each expression. \(\frac{2(-2)-4(-3)}{5-8}\)

Step-by-Step Solution

Answer

-2.67

Step 1: Simplify inside parentheses

There are two operations within the parentheses, multiplication in both cases: \(2(-2) = -4\) and \(-4(-3) = 12\). Replace these in the original problem, so it becomes: \(\frac{-4+12}{5-8}\)

Step 2: Simplify the numerator and denominator separately

The numerator -4+12 is equal to 8, and the denominator 5-8 is equal to -3. This simplifies the expression to: \(\frac{8}{-3}\)

Step 3: Simplify the final fraction

In the final expression \(\frac{8}{-3}\), the fraction can be simplified further by dividing 8 by -3 to get -2.67 (rounded to hundredths place)

Key Concepts

Simplifying Expressions

Simplifying expressions is all about reducing equations to their simplest form while keeping their meaning intact. It makes math much easier to handle and understand. Think of it like cleaning your messy room - we're just tidying up!

• In mathematics, expressions can be rewritten to look simpler, as long as the overall value doesn't change.
• Usually, this involves combining like terms, cutting out unnecessary parts, or performing arithmetic operations.

In our example, we simplify \(\frac{2(-2)-4(-3)}{5-8}\) by working out what's in the parentheses first. So, 2 times -2 becomes -4 and -4 times -3 becomes 12. After this, the expression simplifies to \(\frac{-4+12}{5-8}\). In this simplified form, math becomes a lot more approachable.

Numerator and Denominator

Every fraction consists of two main parts: the numerator on top and the denominator on the bottom. Think of the fraction as a division problem. The numerator is the number you have, and the denominator tells you how many parts to divide it into.

• The numerator is the number above the fraction line, indicating the "part of" you're dealing with.
• The denominator is the number below the line, showing how many equal parts the whole is divided into.

In our expression \(\frac{-4+12}{5-8}\), the numerator is \-4 + 12\ which simplifies to 8. The denominator is \5-8\, simplifying to -3. Understanding this distinction helps make the next step, fraction simplification, much easier.

Fraction Simplification

Simplifying fractions is about making them as straightforward as possible while keeping them equivalent. It’s like cutting a pizza into fewer slices but still having the whole pie.

• A fraction is simplified when both the numerator and the denominator have no divisors in common, other than 1.
• Dividing both parts by the greatest common divisor (GCD) will usually do the trick.

With the expression \(\frac{8}{-3}\), the fraction \(\frac{8}{-3}\) is already in its simplest form because 8 and -3 don’t have any common divisors other than 1. Even though 8 divides by 3 evenly gives -2.67, in proper terms it's simpler to leave the answer in fractional form for precision.

Multiplication

Multiplication involves adding a number to itself a certain number of times. It is one of the basic operations in mathematics and plays a key role in simplifying expressions.

• For example, multiplying 2 by -2 means adding two instances of -2, resulting in -4.
• Similarly, multiplying -4 by -3 gives us 12 as two negatives make a positive.

In our exercise, identifying and performing multiplication in the term \(2(-2)-4(-3)\) is crucial. Once you've done the multiplication, the expression becomes easier to handle as you can then focus on simplifying the results further. Properly performing these basic operations is the first step to tackling more complex expressions.