Problem 25 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ \frac{4+9}{2+3}-\frac{-3^{2} \cdot 3}{5} $$

Step-by-Step Solution

Answer

The evaluated expression is 8.

Step 1: Evaluate the Numerators

First, calculate the numerators of both fractions. For the first fraction, add the numbers: \(4 + 9 = 13\). For the second fraction, calculate the power and product: \(-3^2 = -9\) and \(-9 \times 3 = -27\).

Step 2: Simplify the Denominators

Next, simplify the denominators. For the first fraction, add the numbers: \(2 + 3 = 5\). The denominator of the second fraction is already simplified as \(5\).

Step 3: Form the Fractions

Now that both numerators and denominators have been evaluated, write the fractions as follows: \( \frac{13}{5} \) and \( \frac{-27}{5} \).

Step 4: Subtract the Fractions

Since both fractions have the same denominator, subtract the numerators directly: \( \frac{13}{5} - \frac{-27}{5} = \frac{13 + 27}{5} = \frac{40}{5} \).

Step 5: Simplify the Result

Simplify the fraction obtained in the previous step. Divide numerator by denominator \( \frac{40}{5} = 8 \).

Key Concepts

Fraction Subtraction

Subtracting fractions can seem a bit tricky at first, but with some practice, it becomes straightforward. When dealing with fractions, the key element to pay attention to is the denominator. The denominator tells us the size of the parts we're working with. If both fractions have the same denominator, as in this exercise, the subtraction process is much easier.

Here's how it works: You subtract the numerators (the top numbers of the fractions) and keep the denominator the same. This is because the denominators indicate that you are working with the same size pieces or parts.

Let's illustrate this with an example:

• Suppose we have two fractions: \(\frac{13}{5}\) and \(-\frac{27}{5}\).


• Both have the same denominator (5), which means the subtraction is simply about subtracting the numerators: \(13 - (-27)\).


• Consider the negative sign in front of 27. It flips the operation into addition, so it becomes \(13 + 27 = 40\).
• The result is \(\frac{40}{5}\).

This method works the same way if you encounter fractions when solving equations or when the numerators or denominators are different. But in those cases, you might need to find a common denominator first.

Exponentiation

Exponentiation is a mathematical operation involving numbers or expressions raised to a power. In this operation, the exponent indicates how many times the base is multiplied by itself.

For instance, let's consider the expression \(-3^2\) in our fraction exercise. The exponent is 2, meaning that you multiply the base, \(-3\), by itself:

• \(-3 \times -3 = 9\)

However, notice in the original problem, the exponent is connected directly to only 3, without parentheses around the negative sign. This means that only 3 is squared. Therefore:

• \(-3^2 = -(3 \times 3) = -9\)


• Like this result, check if the negative sign is squared or not due to the presence or absence of parentheses around the base.

Exponentiation is not only essential for handling negative numbers but is widely used in various applications such as calculating area, volume, or even in scientific computations. Understanding how to manage signs and apply exponents will strengthen both algebraic and arithmetic skills.

Simplification Steps

Simplification is all about making mathematical expressions easier to understand and manage. Once you've done operations like addition, subtraction, multiplication, or exponentiation, the result can often look complicated. Simplifying these expressions makes them clearer and more efficient to work with.

Here’s a breakdown of how to simplify the result from our fraction subtraction:

• We arrived at the fraction \(\frac{40}{5}\).


• To simplify it, divide the numerator (40) by the denominator (5).


• This operation results in a whole number: \(8\).


• Simplifying doesn’t end with fractions. Apply similar techniques to simplify expressions with variables or when you encounter square roots or other complex mathematical forms.

Simplification helps in reducing errors and provides a clearer path to solutions when dealing with intricate mathematical problems. Regular practice will improve your ability to spot ways to simplify efficiently.