Problem 95 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{(-4)^{2}-16}{4-12}\)

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Solve the Exponent

First, evaluate the expression inside the numerator's exponent. We have \((-4)^2\). Calculating this, we get:\[(-4)^2 = (-4) \times (-4) = 16\]So now the expression becomes:\[\frac{16 - 16}{4 - 12}\]

Step 2: Subtract in the Numerator

Subtract the numbers in the numerator:\[16 - 16 = 0\]So the expression now looks like this:\[\frac{0}{4 - 12}\]

Step 3: Subtract in the Denominator

Perform the subtraction in the denominator:\[4 - 12 = -8\]Our expression is now:\[\frac{0}{-8}\]

Step 4: Evaluate the Fraction

Finally, compute the fraction:\[\frac{0}{-8} = 0\] because any number divided by a non-zero number is 0.

Key Concepts

Exponents

One of the first steps in evaluating expressions with exponents is to handle any terms involving powers. In our example, we begin by solving \((-4)^2\). Exponents indicate that a number (the base) is multiplied by itself the number of times specified by the power.
In this case, \((-4)^2\) means \(-4\) is multiplied by \(-4\).

• Base: The number being multiplied, here it’s \(-4\).
• Exponent: The number of times the base is used as a factor, here it’s \(2\).

So, \((-4) imes (-4) = 16\). Remember, when multiplying two negative numbers, the result is a positive number. After solving the exponent, the expression simplifies with the result: \(16\).

Numerator Subtraction

The next step in the evaluation process is the subtraction in the numerator. After simplifying the exponent, the expression is \(16 - 16\).
Subtraction in this step involves simply subtracting these two numbers within the numerator:

• Start from the left: Move from the first number to the second using subtraction.
• Result when identical: If both numbers are the same, such as in \(16 - 16\), the result is always \(0\).

So, \(16 - 16 = 0\) is straightforward, leading to a numerator of \(0\).

Denominator Subtraction

Similar to the numerator subtraction, you next handle the subtraction in the denominator, starting with \(4 - 12\).
Subtraction here also requires the same basic principle as in the numerator:

• Order of operation: Begin with the leftmost number, reducing it by the number to the right.
• Negative result: If the second number is larger than the first, the result is negative.

In this case, \(4 - 12\) leads to \(-8\), as \(12\) is greater than \(4\). It is important to handle negative results carefully to ensure accuracy.

Division by Zero Rule

In fraction evaluation, understanding the division by zero rule is crucial for avoiding mistakes. However, in this example, after simplifying the numerator and denominator, we compute \(\frac{0}{-8}\). When the numerator is \(0\) and the denominator is any non-zero number, the fraction evaluates to \(0\).
Key rules include:

• Non-zero Denominator: Division is permitted as long as the denominator is non-zero.
• Zero Numerator: If numerator is zero, the result is always \(0\), regardless of the denominator.
• Division by Zero: Never allow a denominator of \(0\); it makes the expression undefined.

Thus, the final expression \(\frac{0}{-8}\) confirms the result is \(0\). It is crucial to recognize these rules for correct fraction evaluation.