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Chapter 13: Problem 64

Evaluate each of the following. \(\frac{(-1)^{k}}{k-5},\) for \(k=10\)

Short Answer

Expert verified
The evaluated expression is \(\frac{1}{5}\).

Step by step solution

01

Substitute the value of k

First, substitute the given value of 10 in place of k in the expression \(\frac{(-1)^{k}}{k-5}\).
02

Calculate the numerator

Evaluate the numerator \((-1)^{k}\) by substituting \(k=10\). So, \((-1)^{10} = 1\).
03

Calculate the denominator

Determine the denominator of the fraction by substituting \(k=10\): \(k-5 = 10-5 = 5\).
04

Form the fraction

Now, form the fraction with the results from Step 2 and Step 3 which gives us \(\frac{1}{5}\).
05

Simplify the fraction

The fraction \(\frac{1}{5}\) is already in its simplest form. The evaluated expression is \(\frac{1}{5}\).

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Key Concepts

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

Substitution
Substitution is the process of replacing a variable in an expression with a given value. In our exercise, we replaced the variable \(k\) with the value 10.
This is done to evaluate the expression for that specific value.
Let's see how the substitution works:
  • Start with the original expression: \( \frac{(-1)^k}{k-5} \)
  • Substitute 10 for \(k\): \(\frac{(-1)^{10}}{10-5} \)
We now move towards calculating the numerator and denominator.
Exponentiation
Exponentiation involves raising a number (the base) to the power of an exponent. In our problem, we needed to evaluate \((-1)^{10}\).
Remember these rules when dealing with exponents:
  • If the exponent is even, a negative number raised to that power will be positive: \((-1)^{10} = 1\)
  • If the exponent is odd, the result will be negative: \((-1)^9 = -1\)
For our case, since 10 is even, \((-1)^{10} = 1\). This helps us get the numerator.
Simplifying Fractions
Simplifying fractions makes them easier to understand and compare. A fraction is in simplest form when the numerator and denominator have no common factors other than 1.
Our expression was \( \frac{1}{5} \), which is already in its simplest form:
  • The numerator is 1 (no factors other than 1 itself).
  • The denominator is 5 (a prime number, so also has no factors other than 1 and itself).
Thus, the fraction \( \frac{1}{5} \) does not need further simplification.
Numerator and Denominator
Numerators and denominators are essential components of a fraction.
  • The **numerator** is the top part of the fraction and represents how many parts we have. In our case, it was 1.
  • The **denominator** is the bottom part of the fraction and represents the total number of equal parts into which the whole is divided. For us, it was 5.
Understanding their role helps in working with fractions effectively. By substituting the value of 10 for \(k\), we calculated:
  • Numerator: \((-1)^{10} = 1\)
  • Denominator: \(10 - 5 = 5\)
Forming the fraction \( \frac{1}{5} \).

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