Problem 28 - Elementary and Intermediate Algebra

Question

Express using positive exponents and simplify, if possible. \((-8)^{-1}\)

Step-by-Step Solution

Answer

\( (-8)^{-1} = \frac{1}{-8} \)

Step 1: Understanding Negative Exponents

A negative exponent indicates that we take the reciprocal of the base raised to the corresponding positive exponent. So, for \((-8)^{-1}\), we take the reciprocal of \(-8\).

Step 2: Calculating the Reciprocal

To find the reciprocal of \(-8\), we exchange the numerator and the denominator. Since \(-8\) can be written as \(\frac{-8}{1}\), its reciprocal is:\[\frac{1}{-8}.\]

Step 3: Expressing with Positive Exponents

With the negative exponent accounted for, \((-8)^{-1}\) is expressed with positive exponents as:\[\frac{1}{-8}.\]

Key Concepts

Reciprocal

Understanding the concept of reciprocal is crucial, especially when dealing with negative exponents. In simple terms, a reciprocal is what you get when you flip the numerator and the denominator of a fraction. It's like turning something upside down. For example, the reciprocal of 8 is \(\frac{1}{8}\). Reciprocals are used heavily in math to transform negative exponents into positive ones. When a number is a whole number, like -8 in our example, it can be written as a fraction \(\frac{-8}{1}\). Here, switching the numerator with the denominator gives us \(\frac{1}{-8}\). Thus, finding a reciprocal is essential to handling negative exponents.

Positive Exponents

The shift from negative to positive exponents involves using the reciprocal of the base number. A positive exponent indicates how many times a number is multiplied by itself. For example, \(8^2\) means 8 is multiplied by itself, resulting in 64. Positive exponents help to simplify expressions and make calculations easier. When we have a negative exponent, like in \((-8)^{-1}\), its positive counterpart becomes apparent once we convert into the reciprocal: \(\frac{1}{-8}\). This form is simpler to work with and avoids the confusion brought by negative exponents.

Simplifying Expressions

Simplifying expressions means rewriting them in a way that is easier to understand and work with. This often involves reducing fractions or rewriting exponents to make the math manageable. When simplifying expressions involving negative exponents, this usually involves converting them into positive exponents. As seen in \((-8)^{-1} \,which simplifies to \ \frac{1}{-8}\), the expression is presented in its simplest positive-exponent form. Here are a few tips to remember while simplifying:

• Always convert negative exponents into positive by using their reciprocal.
• Combine like terms where possible.
• Check to see if fractions can be simplified further by reducing them into smaller terms.

Simplification is useful in ensuring that we have the simplest and most accurate representation of a mathematical problem.