Problem 38 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(-6^{-3}\)

Step-by-Step Solution

Answer

\(\frac{1}{-216}\)

Step 1: Apply the relevant algebraic rules

Use properties of exponents, radicals, or algebraic identities to simplify the expression.

Step 2: State the simplified result

The simplified expression is \(\frac{1}{-216}\).

Key Concepts

Simplification of Negative Exponents

Negative exponents might look puzzling at first glance, but they simply imply a specific mathematical operation. Rather than thinking of the negative sign as a subtraction, view it as a way of expressing the reciprocal or the inverse. Simplification involves turning expressions with negative exponents into their positive counterparts.

Take the expression \(-6^{-3}\). What this really means is: take \(-6\) and raise it to the power of 3, but take the reciprocal due to the negative exponent. This changes everything!

You write it as:

• \((-6)^{-3} = \frac{1}{(-6)^3}\)

Now you see? No more negative exponents! It's all about flipping the base to the bottom of a fraction, converting the negative exponent into a positive one. This makes it easier to work with and follow up calculations.

Understanding Reciprocal in Algebra

The term "reciprocal" might sound complex, but it's quite straightforward. In mathematics, a reciprocal refers to the "flipping" of a number. Specifically, for any non-zero number \(a\), its reciprocal is \(\frac{1}{a} \).

When dealing with negative exponents, the reciprocal plays a crucial role. It turns the problem on its head – literally – by changing the exponent to positive.

For the expression \(-6^{-3}\), the reciprocal transformation helps us:

• By understanding \(-6^{-3}\) as \(\frac{1}{(-6)^3}\), it becomes evident how the negative exponent translates the power operation into division, rather than a multiplication.

This simple flip clears away any confusion surrounding negative exponents and helps us manage them easily.

Converting to Positive Exponents

Positive exponents indicate straightforward multiplication. They are more intuitive to work with compared to their negative counterparts. Once a number is stripped away from its negative exponent using the reciprocal rule, we're left entirely with positive exponents.

This conversion is vital, as using positive exponents makes calculations straightforward:

• In our case of \(\frac{1}{(-6)^3}\), we compute the power \((-6)^3\), turning the base into a new single positive operation.

The process involves simply multiplying the base by itself as many times as indicated by the positive exponent, without worrying about fractions. This not only simplifies computation but also readability. Once you convert to positive exponents, mathematics feels simpler and more accessible!