Problem 21 - Elementary and Intermediate Algebra

Question

Express using positive exponents and simplify, if possible. \(2^{-2}\)

Step-by-Step Solution

Answer

\(2^{-2} = \frac{1}{4}\).

Step 1: Understanding Negative Exponents

When dealing with negative exponents, remember that a negative exponent indicates that the base should be taken as a reciprocal. For any number with a negative exponent, such as \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\). Thus, \(2^{-2}\) should be rewritten.

Step 2: Rewriting with Positive Exponents

Apply the rule for negative exponents to \(2^{-2}\). Rewrite it as the reciprocal of \(2^2\): \(2^{-2} = \frac{1}{2^2}\). This changes the exponent from negative to positive.

Step 3: Calculating the Power

Now, calculate \(2^2\). Raising 2 to the power of 2 means multiplicatively duplicating the base twice: \(2 \times 2 = 4\). So, \(2^2 = 4\).

Step 4: Simplifying the Expression

Replace \(2^2\) in the denominator with its calculated value to get \(\frac{1}{4}\). This simplifies the expression fully using positive exponents.

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first glance, but they're quite straightforward. A negative exponent tells you to find the reciprocal of the base raised to the absolute value of the given exponent. In simpler terms, for any expression like \(a^{-n}\), you can rewrite it as \(\frac{1}{a^n}\). The negative sign of the exponent simply directs us to flip the base into a fraction.
To illustrate: \(2^{-2}\) converts to \(\frac{1}{2^2}\). Here, the base "2" is raised to the exponent 2, and since it's a negative exponent, the result is expressed as the reciprocal. Understanding this concept will help simplify expressions and solve problems efficiently.

Positive Exponents

Positive exponents denote straightforward multiplication of the base. An expression like \(a^n\) means multiplying the base 'a' by itself 'n' times. This is intuitive as it's simply about counting how many times the base is used as a factor.
For example, \(2^2\) is evaluated as \(2 \times 2\), which equals 4. Positive exponents will always produce increasingly larger values as the base is multiplied more times, assuming the base is greater than one. The change from negative to positive can simplify calculations and help us find clear results.

Reciprocal

The term reciprocal refers to flipping a fraction, essentially swapping the numerator and the denominator. When dealing with whole numbers, the reciprocal of a number is simply 1 divided by that number.
For instance, the reciprocal of 4 is \(\frac{1}{4}\). When negative exponents are involved, understanding reciprocals helps us transition expressions into a form using positive exponents.

• Reciprocal of \(a\) is \(\frac{1}{a}\)
• Reciprocal of \(\frac{b}{c}\) is \(\frac{c}{b}\)

Knowing how to find the reciprocal is crucial when simplifying expressions involving negative exponents, ensuring clarity and precision in calculations.