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Chapter 10: Problem 420

(a) Explain the meaning of the exponent in the expression \(2^{3}\) . (b) Explain the meaning of the exponent in the expression \(2^{-3}\)

Short Answer

Expert verified
\(2^3\) means 2 multiplied by itself 3 times and equals 8. \(2^{-3}\) means \(\frac{1}{2^3}\), which equals \(\frac{1}{8}\).

Step by step solution

01

Title - Understanding Positive Exponents

The exponent in an expression like \(2^3\) tells us how many times the base number, which is 2, is multiplied by itself. In this case, \(2^3\) means \(2 \times 2 \times 2\), which equals 8.
02

Title - Understanding Negative Exponents

The exponent in an expression like \(2^{-3}\) indicates the reciprocal of the base raised to the positive exponent. Thus, \(2^{-3}\) means \(\frac{1}{2^3}\). Compute \(2^3\) first, which is 8, so \(2^{-3} = \frac{1}{8}\).
03

Title - Summarizing Results

For \(2^3\), the result is 8, because 2 is multiplied by itself 3 times. For \(2^{-3}\), the result is \(\frac{1}{8}\), because it is the reciprocal of \(2^3\).

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Key Concepts

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

Understanding Positive Exponents
In mathematics, exponents are a way to express repeated multiplication of a number by itself. A positive exponent indicates how many times the base number is being multiplied. For example, in the expression \(2^3\), the base is 2 and the exponent is 3. This means that you multiply 2 by itself three times:
\[2^3 = 2 \times 2 \times 2 = 8\]
This is straightforward: the exponent tells you the number of times to use the base in a multiplication. Positive exponents always result in a larger number (assuming the base is greater than 1).
Understanding Negative Exponents
Negative exponents might seem tricky at first, but they follow a simple rule: a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In simpler terms, if you have \(2^{-3}\), it means:
\[2^{-3} = \frac{1}{2^3}\]
Now, compute \(2^3\) first, which is 8:
\(2^3 = 8\)
So, \(2^{-3}\) becomes:
\[2^{-3} = \frac{1}{8}\]
Whenever you see a negative exponent, remember it鈥檚 about creating a fraction where you put 1 over the base raised to the positive version of that exponent.
Understanding Reciprocals
A reciprocal is simply the inverse of a number. In the context of exponents, if you have an expression with a negative exponent, you鈥檙e dealing with a reciprocal. For example, the reciprocal of 2 is \(\frac{1}{2}\)
To convert a number to its reciprocal, you flip it. If the number is a fraction, you switch the numerator and the denominator.
For instance, the reciprocal of \(\frac{1}{2}\) is 2, and the reciprocal of 5 is \(\frac{1}{5}\).
So, when we handled \(2^{-3}\) and found it to be \(\frac{1}{8}\), we used the concept of reciprocals. Whenever you encounter negative exponents, think of flipping the base and working with its positive exponent.

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