Problem 83 - Elementary Algebra

Question

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{5}{81-k^{2}} $$

Step-by-Step Solution

Answer

The expression is undefined for \( k = 9 \) and \( k = -9 \).

Step 1: Identify when the expression is undefined

A rational expression is undefined when the denominator equals zero. Therefore, find the values of the variable that make the denominator zero.

Step 2: Set the denominator equal to zero

Write the equation: \[ 81 - k^2 = 0 \]

Step 3: Solve for the variable

Rearrange the equation to solve for \( k \). This can be done by factoring or using the difference of squares formula: \[ 81 - k^2 = (9 + k)(9 - k) = 0 \]

Step 4: Find the values of the variable

Set each factor equal to zero and solve for \( k \): \[ 9 + k = 0 \Rightarrow k = -9 \] \[ 9 - k = 0 \Rightarrow k = 9 \]

Step 5: Conclude the solution

The values of the variable that make the expression undefined are \( k = 9 \) and \( k = -9 \).

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In simpler terms, it is a fraction that has variables in the denominator.
For example, the rational expression \(\frac{2x + 3}{x - 5}\) has a numerator \(2x + 3\) and a denominator \(x - 5\). This means it's a proper fraction, but with variables involved.
To understand rational expressions, you need to know:

• How to identify the numerator and the denominator
• The common rules of fractions
• Special rules for algebraic fractions

It is important to realize that a rational expression is undefined when its denominator equals zero. Let's explore this in more detail next.

Denominator Equals Zero

In rational expressions, it is crucial to remember that the expression becomes undefined if the denominator equals zero. This is because division by zero is not possible in mathematics.
Consider the expression \(\frac{a}{b}\). For any real number 'a', this expression is undefined if 'b' equals zero.
In the given problem, the rational expression \(\frac{5}{81 - k^2}\) will be undefined when the denominator \((81 - k^2)\) equals zero. Let's look into how we can solve this specific problem:
Step 1: Set the denominator equal to zero: \ 81 - k^2 = 0 \.
Step 2: Therefore, the values of k that make the expression undefined are the solutions to the equation \ 81 - k^2 = 0\. To solve this, we use the difference of squares.

Difference of Squares

The difference of squares is a specific algebraic technique used to simplify expressions like \(81 - k^2\). This is a special case because it can be factored into two binomials.
The general formula for the difference of squares is \( a^2 - b^2 = (a + b)(a - b)\).
Applying this to the given expression \ 81 - k^2 \:
We notice \(a = 9\) and \(b = k\) based on the equation structure. So, \ 81 - k^2 = (9 + k)(9 - k) = 0.\
This breaks down into two simpler equations: \ 9 + k = 0 \ and \ 9 - k = 0 \.
Solving these, we get:

• \ 9 + k = 0 \rightarrow k = -9\
• \ 9 - k = 0 \rightarrow k = 9 \

Therefore, the expression \ \frac{5}{81 - k^2} \ is undefined when k equals 9 or -9.