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Chapter 0: Problem 13

In Exercises \(1-28\), compute the numbers. $$ 6^{-1} $$

Short Answer

Expert verified
\( 6^{-1} = \frac{1}{6} \)

Step by step solution

01

Understand the Negative Exponent

The notation for a number raised to a negative exponent means to take the reciprocal of the number raised to the corresponding positive exponent. Here, we need to compute the reciprocal of the number raised to the power of 1.
02

Reciprocal Definition

The reciprocal of a number is defined as \( \frac{1}{\text{number}} \).
03

Apply the Reciprocal

For \( 6^{-1} \), we take the reciprocal of 6: \( 6^{-1} = \frac{1}{6} \).

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Key Concepts

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

Reciprocal
Let's understand what a reciprocal is. The reciprocal of a number is simply \(\frac{1}{\text{number}}\). For example, the reciprocal of 2 is \(\frac{1}{2}\), and the reciprocal of 5 is \(\frac{1}{5}\). When you take the reciprocal of a number, you are essentially flipping it over. This concept is very important when dealing with negative exponents. Remember, if you have a fraction, such as \(\frac{2}{3}\), its reciprocal would be \(\frac{3}{2}\). Reciprocals help simplify computations, especially when handling exponents.
Negative Exponent
A negative exponent means we are taking the reciprocal of the number with a positive exponent. For instance, for \(6^{-1}\), it translates to \(\frac{1}{6}\). This means that instead of multiplying 6 by itself, we take the reciprocal. Another example is \(2^{-3}\). This does not just mean that we invert the 2; we also raise it to the power of 3 and then take the reciprocal: \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\). As you see, the rule is consistent: negate the exponent, convert it to a positive one and then take the reciprocal of the base raised to that power.
Power of a Number
When dealing with powers of a number, you are multiplying that number by itself as many times as specified by the exponent. For example, \(5^3\) means 5 multiplied by itself three times: \(5 \times 5 \times 5 = 125\). Keep in mind that if you face a negative exponent, such as in \(3^{-2}\), you first convert it to the reciprocal format, like \(\frac{1}{3^2}\). Thus, \(3^{-2}\) equates to \(\frac{1}{9}\). Notice how understanding powers, reciprocals, and negative exponents interplay with each other. This understanding will significantly bolster your proficiency in math.

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Most popular questions from this chapter

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\sqrt[3]{x} \cdot \sqrt[3]{x^{2}}\)

A specialty shop prints custom slogans and designs on T-shirts. The shop's total cost at a daily sales level of \(x\) T-shirts is \(C(x)=73+4 x\) dollars. (a) At what sales level will the cost be \(\$ 225\) ? (b) If the sales level is at 40 T-shirts, how much will the cost rise if the sales level changes to 50 T-shirts?

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