Problem 11 - College Algebra

Question

Evaluate each exponential expression. $$ 4^{-3} $$

Step-by-Step Solution

Answer

The result of the evaluation of the exponential expression \(4^{-3}\) is \(1/64\).

Step 1: Identify the base and the exponent

The base in the expression \(4^{-3}\) is 4, and the exponent is -3.

Step 2: Treat the negative exponent

Negative exponents indicate that the reciprocal of the base should be used. The reciprocal of 4 is 1/4. So now, the expression is \((1/4)^3\).

Step 3: Evaluate the expression

Finally, taking the absolute value of exponent 3, cube 1/4, which results in \(1/64\).

Key Concepts

Negative Exponents

Negative exponents are a key concept in mathematics that often puzzle students. They denote a special operation performed on numbers, indicating that the base should be inverted, or made into its reciprocal. Instead of performing traditional multiplication as you would with positive exponents, a negative exponent tells us to "flip" the number. For instance, in the expression \( 4^{-3} \), the negative exponent \(-3\) requires us to find the reciprocal of \(4\).

In practice, this means converting the base \(4\) into \(\frac{1}{4}\) and changing the sign of the exponent to positive as \((\frac{1}{4})^{3}\). This action of inverting the number is what transforms the negative exponent into a positive operation. By grasping the meaning of negative exponents, students can simplify many mathematical problems with ease.

Base and Exponent

Understanding the components of an exponential expression is crucial for solving these types of equations. The base is the number that gets multiplied, and the exponent tells us how many times to multiply the base by itself.

Consider the expression \(4^{-3}\). Here, the base is \(4\), which is the number we work with, and the exponent is \(-3\), which indicates how we transform that base. Rather than multiplying \(4\) by itself, the negative exponent tells us to find the reciprocal first.

Recognizing the base and exponent not only aids in evaluating expressions but also helps in understanding their transformation when dealing with operations like reciprocals and powers.

Reciprocals

Reciprocals are an essential part of working with negative exponents. A reciprocal essentially turns a number inside out. The number \(4\) becomes its reciprocal, \(\frac{1}{4}\). It's like thinking of \(4\) as \(\frac{4}{1}\) and flipping it.

When you encounter a negative exponent, as in \(4^{-3}\), the first step is finding this reciprocal, transforming the base to \(\frac{1}{4}\).

• Reciprocals simplify division: any number times its reciprocal equals 1.
• They help turn negative exponent problems into manageable tasks.

By converting the base into a reciprocal, you convert the problem into evaluating a positive exponent, simplifying the process greatly.

Evaluating Powers

Once the base is rewritten with a positive exponent, evaluating the power becomes straightforward. In the expression \((\frac{1}{4})^{3}\), this becomes equivalent to multiplying \(\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4}\).

Here’s how you can evaluate it step-by-step:

• Multiply the reciprocal base: \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\).
• Then, multiply again by another \(\frac{1}{4}\): \(\frac{1}{16} \times \frac{1}{4} = \frac{1}{64}\).

So, the calculation gives us \(\frac{1}{64}\). Evaluating powers is about systematic multiplication, which becomes much easier after grasping the conversion using negative exponents. By repeatedly multiplying the base, students can master powers efficiently.