Problem 92 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{6+(-2)^{2}}{4-9} $$

Step-by-Step Solution

Answer

The simplified expression is -2.

Step 1: Evaluate the Exponent

Begin by evaluating the exponent within the numerator. The expression inside the parenthesis is \((-2)^2\). When a negative number is squared, it becomes positive: \((-2)^2 = 4\). Thus, the numerator becomes \(6 + 4\).

Step 2: Simplify the Numerator

Now, simplify the numerator by adding the numbers: \(6 + 4 = 10\). So, the numerator is simplified to \(10\).

Step 3: Simplify the Denominator

Next, simplify the denominator \(4 - 9\). Subtract 9 from 4: \(4 - 9 = -5\). Thus, the denominator is \(-5\).

Step 4: Division of Numerator by Denominator

Divide the simplified numerator by the simplified denominator: \(\frac{10}{-5}\). Dividing 10 by -5 results in \(-2\).

Step 5: Final Simplified Answer

Therefore, the simplified expression of \(\frac{6+(-2)^{2}}{4-9}\) is \(-2\).

Key Concepts

Understanding Exponents

When you see an exponent, it's a way of indicating that a number, called the base, is multiplied by itself a certain number of times. In

• \((-2)^2\)

this expression,

• \(-2\) is the base, and
• 2 is the exponent.

This means:

• \(-2 \times -2 = 4\).

It's crucial to remember that any negative number raised to an even power (like square, i.e., raised to 2) will result in a positive value.So, in our example, even though we start with a negative base, the power is 2, which neutralizes the negative, making the result positive:

• \((-2)^2 = 4\).

Exponents are often used for simplifying repeated multiplication into a more concise form. This simplification helps evaluate complex expressions accurately.

Numerator and Denominator in Fractions

Fractions consist of two main parts: the numerator and the denominator. Understanding how they work is key to simplifying expressions. Let's break down their functions:

• The numerator is the top part of the fraction and represents how many parts of the whole are being considered.
• The denominator, on the other hand, is the bottom part and represents the total number of equal parts in the whole.

For the fraction

• \(\frac{10}{-5}\),

we have:

• 10 as the numerator and
• -5 as the denominator.

The simplification process involves evaluating these two parts independently first. In this example:1. We calculated the numerator by adding

• 6 and 4 to obtain 10.

2. For the denominator, subtracting

• 9 from 4 gave us \(-5\).

By understanding and simplifying each part, we ensure accurate simplification of the full fractional expression.

Simplifying Division of Integers

Division of integers involves finding out how many times one integer (the denominator) fits into another (the numerator). This operation is straightforward with positive numbers, but can be tricky with negatives. Our exercise involves

• division of integers where both the numerator (10) and the denominator (-5) are involved,

resulting in:

• \(\frac{10}{-5} = -2\).

Here's a straightforward method:- Divide the absolute values of the integers:

• 10 divided by 5 gives 2.

- Determine the sign:

• If one of the numbers is negative (like -5), the result is negative.

In general, multiplying or dividing integers follows a specific rule regarding signs:

• Positive \(\times\) Positive = Positive
• Negative \(\times\) Negative = Positive
• Positive \(\times\) Negative = Negative

This rule ensures that you can evaluate similar problems in future exercises with greater confidence and ease.