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Chapter 1: Problem 66

Evaluate. If an expression is undefined, say so. $$\frac{4^{2}}{-4^{2}-(-16)}$$

Short Answer

Expert verified
The expression is undefined because the denominator is zero.

Step by step solution

01

- Evaluate the Numerator

First, evaluate the numerator of the given expression. The numerator is given by 4^{2}. Calculate this value: \[ 4^{2} = 16 \]
02

- Simplify the Denominator

Next, simplify the denominator of the given expression. The denominator is \(-4^{2} - (-16)\). First, evaluate \(-4^{2}\): \[ -4^{2} = -16 \] Next, simplify -(-16): \[ -(-16) = 16 \]
03

- Combine the Denominator Values

Combine the simplified parts in the denominator: \[ -16 + 16 = 0 \]
04

- Determine if the Expression is Defined

Now, combine the values from the numerator and the denominator into the expression: \[ \frac{16}{0} \] Remember, any number divided by zero is undefined.

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Key Concepts

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

numerator
In an algebraic fraction, the numerator is the top part or the 'top number' of the fraction. It represents how many parts of a whole you have. For the given exercise \(\frac{4^{2}}{-4^{2}-(-16)}\), the numerator is \(4^{2}\). Evaluating this, \(4^{2} = 16\). This simplifies the initial expression to \(\frac{16}{\text{denominator}}\). When calculating a fraction, always start by simplifying the numerator.
denominator
The denominator is the bottom part of a fraction that shows into how many parts the whole is divided. For the given expression, the denominator is \(-4^{2} - (-16)\). Start by evaluating \(-4^{2} = -16\). Then, simplify \(-(-16)\), which means turning \(-16\) positive to get \(16\). This makes the denominator: \(-16 + 16 = 0\). Always pay attention to the signs in the denominator when simplifying.
undefined expressions
An undefined expression occurs when the denominator of a fraction equals zero. For example, in this exercise, after simplification, the denominator becomes zero: \(\frac{16}{0}\). Division by zero is undefined in mathematics because it does not result in a finite or meaningful number. When you encounter \(\frac{a}{0}\), the result is always considered undefined. Hence, it is crucial to check if your denominator ends up being zero to determine if the expression is undefined.
simplification
Simplification is the process of reducing an algebraic expression to its simplest form. It involves combining like terms, evaluating powers, and addressing any arithmetic operations within the expression. For instance, in this exercise, we simplified both the numerator and the denominator before determining the overall value or state of the fraction. For the numerator: \(4^{2} = 16\), and the denominator led to zero: \(-4^{2} - (-16) = 0\). This practice helps in identifying the nature of the expression, whether it is defined, undefined, or simplified to a more manageable form.

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