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Chapter 14: Problem 4

Give the value of each of the following. $$\left(5^{-1}\right)^{-1}$$

Short Answer

Expert verified
The value of \( \left(5^{-1}\right)^{-1} \) is 5.

Step by step solution

01

Simplify the Inner Exponent

First, simplify the expression within the brackets using the negative exponent rule. The expression \(5^{-1}\) equals \(1 / 5\). This results in \(\left(1/5\right)^{-1}\).
02

Simplify the Outer Exponent

Next, deal with the remaining negative exponent. Apply the negative exponent rule to simplify \( (1/5)^{-1}\) which equals \(1 / (1/5)\).
03

Simplify the Fraction

In dividing by a fraction, multiply by its reciprocal. That means \(1 / (1/5)\) is equivalent to \(1 * 5/1\).
04

Final Simplification

Finally, simplify the fraction \(5/1\) to get the final answer. This results in 5.

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Key Concepts

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

Understanding the Reciprocal
The concept of the reciprocal is crucial when dealing with negative exponents. The reciprocal of a number is basically one divided by that number. For example, the reciprocal of 5 is \(\frac{1}{5}\). It flips the fraction upside down.

When you see a negative exponent, it often signals the reciprocal is involved. In the exercise, the inner term \(5^{-1}\) means we need its reciprocal, which is \(\frac{1}{5}\).

This is why \(5^{-1}\) becomes \(\frac{1}{5}\). The negative exponent flips the base of the original term, making it a reciprocal.
Mastering Fraction Simplification
Simplifying fractions is all about making expressions easier to understand. It involves rearranging the fraction so it's expressed most simply.

In the exercise, after turning \(5^{-1}\) into \(\frac{1}{5}\), we simplify further by dealing with \((\frac{1}{5})^{-1}\).

Using the reciprocal here means multiplying \(1\) by \(\frac{5}{1}\). This simplifies to \(5/1\), or just 5.

Remember, dividing by a fraction is the same as multiplying by its reciprocal. This is a handy technique to simplify fractions effectively.
Navigating Exponent Rules
Exponent rules are essential for handling expressions with powers. The key rule with negative exponents, for instance, tells us that a negative exponent means taking the reciprocal.

For example, \(a^{-n}\) turns into \(\frac{1}{a^n}\). In our exercise, \(5^{-1}\) transforms to \(\frac{1}{5}\) because of this rule. Then, \((\frac{1}{5})^{-1}\) becomes \(5/1\) due to the reciprocal's influence.

Understanding these rules makes manipulating expressions with powers more straightforward. Recognizing these patterns makes solving such exercises easier and enhances mathematical flexibility.

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Most popular questions from this chapter

For each transformation given in Exercises \(5-10\) : a. Plot the three points \(A(0,4), B(4,6),\) and \(C(2,0)\) and their images \(A^{\prime}, B^{\prime},\) and \(C^{\prime}\) under the transformation. b. State whether the transformation appears to be an isometry. c. Find the preimage of \((12,6)\) $$D:(x, y) \rightarrow(3 x, 3 y)$$

If \(h(x)=6 x+1,\) find \(h\left(\frac{1}{2}\right) .\) Is \(h\) a one-to-one function?

The line with equation \(y=x+5\) is reflected in the \(x\) -axis. Find an equation of the image line.

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. \(O\) is the center of the Earth and a point \(P\) of the globe is projected along \(\overrightarrow{O P}\) to a point \(P^{\prime}\) of the cylinder. Is the image of the Arctic Circle congruent to the image of the equator?

Give the value of each of the following. $$9^{-1}$$
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