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Chapter 5: Problem 111

look alikes a. \(6^{-1}\) b. \(-6^{-1}\) c. \(-(-6)^{-1}\)

Short Answer

Expert verified
a. \(\frac{1}{6}\), b. \(-\frac{1}{6}\), c. \(\frac{1}{6}\)

Step by step solution

01

Evaluate 6 to the Power of -1

The expression \(6^{-1}\) means the reciprocal of 6. To calculate this, you take the reciprocal of 6, which is \(\frac{1}{6}\). Thus, \(6^{-1} = \frac{1}{6}\).
02

Evaluate Negative Power on an Expression

For the given expression \(-6^{-1}\), you place a negative sign in front of the reciprocal. The calculation steps are: \(6^{-1} = \frac{1}{6}\). Adding the negative sign yields: \(-6^{-1} = -\frac{1}{6}\).
03

Simplify Double Negative Expression

The expression \(-(-6)^{-1}\) involves both a negative and reciprocal. First, find \((-6)^{-1}\), which is \(-\frac{1}{6}\) because the reciprocal of -6 is \(-\frac{1}{6}\). Next, the outer negative changes \(-\frac{1}{6}\) to \(+\frac{1}{6}\). Therefore, \(-(-6)^{-1} = \frac{1}{6}\).

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Key Concepts

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

Reciprocal
The concept of a reciprocal is foundational in mathematics. It involves flipping a number to represent its inverse. Simply put, the reciprocal of any number is created by swapping the numerator and the denominator.
  • For a whole number like 6, the reciprocal is changing the number into a fraction: from 6 to \(\frac{1}{6}\).
  • The reciprocal essentially inverts the number's value, turning positive integers into fractions under one.
  • Recognize when you have a reciprocal by looking at negative exponents. For instance, \(6^{-1}\) implies the reciprocal, leading directly to \(\frac{1}{6}\).
This understanding helps when simplifying and transforming different math expressions, making calculations easier to handle.
Simplifying Expressions
Simplifying mathematical expressions is all about making them easier to interpret and use. When you face complex-looking expressions, break them down step by step.
  • For expressions like \(-6^{-1}\), first recognize the implication of each component: the \(-1\) exponent signifies a reciprocal.
  • Execute the reciprocal step first, transforming 6 into \(\frac{1}{6}\).
  • The negative sign in front further modifies the expression鈥檚 value, resulting in \(-\frac{1}{6}\).
This careful approach ensures you are simplifying the expression correctly, keeping track of all signs and operations.
Negative Numbers in Exponents
Negative numbers as exponents can appear intimidating at first, but they have a straightforward rule: they signal the reciprocal of the base number.
  • When you see \(6^{-1}\), the \(-1\) exponent tells you to take the reciprocal of 6, forming \(\frac{1}{6}\).
  • With expressions like \(-(-6)^{-1}\), handle negatives in a stepwise approach: flip the base for the reciprocal, then apply any negative signs.
  • Remember, two negatives cancel each other out, so \(-(-6)^{-1}\) = \(-\frac{1}{6}\) becomes \(\frac{1}{6}\).
By following these guidelines, mastering negative exponents becomes manageable, allowing you to simplify expressions accurately.

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Most popular questions from this chapter

The number of feet that a car travels before stopping depends on the driver's reaction time and the braking distance. For one driver, the stopping distance \(d(v),\) in feet, is given by the polynomial function \(d(v)=0.04 v^{2}+0.9 v\) where \(v\) is the velocity of the car in mph. Find the stopping distance at \(60 \mathrm{mph}\). (PICTURE NOT COPY)

Look Alikes \(. .\) \(*\) a. \(\left(2 x^{2}+5\right)+\left(3 x^{2}+2 x+1\right)\) b. \(\left(2 x^{2}+5\right)\left(3 x^{2}+2 x+1\right)\)

Explain what is wrong with the following solution. $$ \text { Solve: } \quad x^{2}-x=6 $$ $$ \begin{array}{l} x(x-1)=6 \\ x=6 \text { or } x-1=6 \end{array} $$ $$ x=-7 $$

Multiply. Assume \(n\) is a natural number. $$ \text { If } f(x)=x^{3}+x, \text { find } f(a+h)-f(a) $$

Aluminum Foil. Find the number of square feet of aluminum foil on a roll if it has dimensions of \(8 \frac{1}{3}\) yards \(\times 12\) inches.
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