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Chapter 1: Problem 95

Simplify each expression. $$\frac{4+(-12)}{(-2)^{2}-4}$$

Short Answer

Expert verified
The simplified result of the expression is undefined.

Step by step solution

01

Simplify the Numerator

Simplify the expression in the numerator \(4+(-12)\) by performing the addition (considering negative sign). The result is -8.
02

Simplify the Denominator

Next, simplify the expression in the denominator \((-2)^{2}-4\) by first performing the exponentiation, then the subtraction. This results in \((4)-4\), which is equal to 0.
03

Perform the Division

Now, using the results from steps 1 and 2, you'll attempt to perform the division -8/0. However, division by zero is undefined in mathematics, therefore the expression cannot be simplified further. The result is 'undefined'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerator Simplification
When simplifying expressions, the numerator is the number or expression above the fraction bar. In our example, the numerator is given by \(4 + (-12)\). Simplifying anything involves breaking it down to its most basic form. Here, this means combining the constant values through basic arithmetic operations like addition or subtraction.

Let's look at the expression \(4 + (-12)\). The plus sign in front of \((-12)\) might be perplexing, but it simply involves adding a negative number. This is the same as subtracting that number:
  • First, recognize \(4 + (-12)\) as \(4 - 12\).
  • Then, carry out the subtraction: \(4 - 12 = -8\).
The simplified numerator is \(-8\). Successfully simplifying the numerator paves the way for the subsequent steps in solving the fraction.
Denominator Simplification
The expression at the bottom of the fraction is called the denominator. It must be simplified to easily evaluate the whole fraction. In this example, the denominator is \((-2)^2 - 4\). Simplifying it involves separate operations you perform one step at a time.

Breaking it down:
  • First, deal with the exponent. Squaring \(-2\) means multiplying \(-2\) by itself, resulting in \(4\) since negative multiplied by negative gives a positive.
  • Then, subtract \(4\) from the square you just computed: \(4 - 4 = 0\).
This results in the denominator being zero.

Simplifying the denominator to zero is pivotal as it changes the way the problem is handled, leading to crucial insights in mathematical practices.
Division by Zero
Division by zero is a well-known concept in mathematics, often emphasizing critical errors to avoid. Simply put, you cannot divide by zero as it doesn鈥檛 render any meaningful or consistent result. When you face a zero in the denominator, like in this example where we end up with \(-8/0\), you encounter an undefined expression.

The foundational rules of arithmetic stand that dividing by zero is impossible, resulting in undefined terms. The consequences include:
  • No value can satisfy the division, as splitting something into zero parts doesn鈥檛 logically follow.
  • Mathematicians establish that expressions like \(-8/0\) have no valid solution and are deemed undefined.
  • The inability to divide by zero also often signals a need to revisit the problem to see any miscalculations or alternate expressions.
This concept reminds us of the careful watch needed when handling divisions in mathematical problems, reaffirming the boundary lines set in mathematical operations.

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