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Chapter 13: Problem 63

Evaluate each of the following. \(\frac{(-1)^{k}}{k-6},\) for \(k=7\)

Short Answer

Expert verified
\(-1\)

Step by step solution

01

Substitute the given value of k

Substitute the given value of \(k\) into the expression. Here \(k = 7\). So, the expression becomes \(\frac{(-1)^{7}}{7-6}\).
02

Simplify the exponent

Calculate \((-1)^{7}\). Since 7 is an odd number, \((-1)^{7} = -1\). The expression now is \(\frac{-1}{7-6}\).
03

Simplify the denominator

Calculate \(7 - 6\). This gives us 1. The expression is now \(\frac{-1}{1}\).
04

Evaluate the fraction

Simplify the fraction \(\frac{-1}{1}\) which is equal to \(-1\).

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Key Concepts

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

substitution
Substitution is a fundamental algebraic technique used to replace a variable with a given value. It helps in evaluating expressions. In our exercise, we were given the expression \(\frac{(-1)^{k}}{k-6}\) and a specific value for \(k = 7\). By substituting \(k\) with \(7\), the expression became \(\frac{(-1)^{7}}{7-6}\). This step is crucial, as it replaces a variable with a known number, making the problem easier to handle.
Some tips to remember:
  • Always verify the value you are substituting.
  • Double-check that you substitute the correct variable.
  • Ensure you rewrite the entire expression with this new value.
exponents
Exponents represent repeated multiplication of a number by itself. In our case, \(\(-1\)^{7}\), the base is -1 and the exponent is 7. Since the exponent (7) is an odd number, our calculation \(\(-1\)^{7}\) results in -1.
Here are some key points about exponents:
  • If the base is -1 and the exponent is odd, the result is -1.
  • If the base is -1 and the exponent is even, the result is 1.
  • Exponents can simplify large calculations.
Understanding these principles helps in evaluating and simplifying expressions involving exponents.
simplification
Simplification involves making an expression easier to understand and solve. After substitution and handling exponents, our next step was simplifying the denominator. From \(\frac{(-1)^{7}}{7-6}\), we then simplified \(7 - 6\) to 1. This step reduced the fraction to \(\frac{-1}{1}\).
Key tips for simplification:
  • Simplify exponents and perform any arithmetic operations next.
  • Always work step-by-step to avoid mistakes.
  • Combine like terms where possible to make calculations easier.
These practices help break down complex expressions into simple, manageable forms.
fractions
Fractions represent a part of a whole. They consist of a numerator and denominator. In this exercise, we needed to simplify \(\frac{-1}{1}\). A fraction can often be simplified further by dividing the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD of \(1\) and \(-1\) is 1, simplifying \(\frac{-1}{1}\) to \(-1\).
Important points about fractions:
  • Always check if you can reduce the fraction further.
  • Understand that a fraction with a denominator of 1 simply represents the numerator.
  • Negative signs can be managed by transferring them between the numerator and the denominator.
By understanding these aspects, one can handle and simplify fractions in algebraic expressions more confidently.

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Most popular questions from this chapter

For the following equations of hyperbolas, complete the square, if necessary, and write in standard form. Find the center, the vertices, and the asymptotes. Then graph the hyperbola. \(4 y^{2}-25 x^{2}-8 y-100 x-196=0\)

To prepare for Section \(13.2,\) review solving quadratic equations (Section \(11.1)\) Solve for \(x\) or for $y . $$\frac{(x-1)^{2}}{25}=1$$

If, in $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ \(a=b,\) what are the asymptotes of the graph? Why?

Find the center and the radius of each circle. Then graph the circle. $$ x^{2}+y^{2}-8 x+2 y+13=0 $$

To prepare for Section \(13.2,\) review solving quadratic equations (Section \(11.1)\) Solve for \(x\) or for $y . $$\frac{1}{9}+\frac{(x-2)^{2}}{4}=1$$
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