Problem 89 - College Algebra Essentials

Question

In Exercises \(83-90\), evaluate each expression without using a calculator. $$32^{-\frac{4}{5}}$$

Step-by-Step Solution

Answer

\(\frac{1}{16}\)

Step 1: Recognize the Negative Exponent

A negative exponent means to take the reciprocal of the base. So, \(32^{-\frac{4}{5}}\) is equivalent to \(\frac{1}{32^{\frac{4}{5}}}\).

Step 2: Exponentiation of 32 to Fraction 4/5

A fractional exponent \(\frac{4}{5}\) on the base \(32\) means that we take the 5th root of \(32\), and then raise the result to the power of 4. We calculate the 5th root of 32 first, which is 2, as \(2^5 = 32\). Then, \(2^4 = 16\). So, \(32^{\frac{4}{5}} = 16\).

Step 3: Calculate the Reciprocal

Since we got \(32^{\frac{4}{5}} = 16\), we can substitute it back into the expression \(\frac{1}{32^{\frac{4}{5}}}\), which becomes, \(\frac{1}{16}\).

Key Concepts

Evaluating Expressions Without Calculator

When tasked with evaluating mathematical expressions without a calculator, understanding the underlying principles can make the process much simpler. First and foremost, familiarize yourself with the order of operations, sometimes remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This guides you through which parts of the expression to solve first.

Specifically, dealing with exponents can seem daunting without a calculator. However, knowing how to handle different types of exponents, including negative and fractional ones, can actually turn this into an approachable task. For example, with the expression \(32^{-\frac{4}{5}}\), you can break it down into smaller parts and progressively simplifying each part. Start by understanding negative and fractional exponents, and then deal with them step by step, applying your knowledge about the base number's properties, such as its prime factors or perfect squares.

Fractional Exponents

Fractional exponents present a unique challenge, but once you understand the principle behind them, they become much less intimidating. Essentially, a fractional exponent like \(a^{\frac{m}{n}}\) can be seen as a two-step process: taking the nth root of the base 'a' and then raising the result to the mth power, or vice versa.

For instance, with \(32^{\frac{4}{5}}\), you would first determine the 5th root of 32, which is \(2\), because \(2^5 = 32\). Then, you take that result and raise it to the 4th power, getting \(2^4 = 16\). It can be helpful to break down the expression into these two main actions and tackle them one at a time to minimize confusion.

Exponentiation Steps

When handling exponents, it's crucial to follow a series of logical steps to simplify the expression correctly. In our example, \(32^{-\frac{4}{5}}\), these steps break down as follows:

Firstly, recognize the type of exponent you are dealing with. If it is negative, like in our example, you know that this entails taking the reciprocal of the base. If the exponent is fractional, you will be performing both a root and a power operation.

Secondly, compute the root implied by the denominator of the fractional exponent. In this case, find the 5th root of 32.

Lastly, raise the result to the power indicated by the numerator. Completing these exponentiation steps in order, as the original solution shows, will lead you to the correct simplified form of the expression. Remember to maintain the proper sequence to avoid missteps.

Reciprocal of a Base

Understanding the reciprocal of a base is a fundamental concept in dealing with negative exponents. The reciprocal of a number is simply 1 divided by that number. Mathematically, if you have a base 'b', the reciprocal is \(\frac{1}{b}\). So for a negative exponent like \(b^{-n}\), you'd flip the base to its reciprocal and apply the positive exponent \(n\) to this reciprocal.

In the context of our example, \(32^{-\frac{4}{5}}\) becomes \(\frac{1}{32^{\frac{4}{5}}}\). After computing \(32^{\frac{4}{5}}\) and finding that it equals 16, the reciprocal of 16, or \(\frac{1}{16}\), provides the final answer. This relationship between a base and its reciprocal is always true, regardless of the number, and is a powerful tool when evaluating expressions with negative exponents.