Problem 26 - Algebra for College Students

Question

Evaluate each numerical expression. \(32^{-\frac{4}{5}}\)

Step-by-Step Solution

Answer

The value of \(32^{-\frac{4}{5}}\) is \(\frac{1}{16}\).

Step 1: Understand Negative Exponents

A negative exponent indicates that the base is on the opposite side of the fraction. Thus, to simplify \(32^{-\frac{4}{5}}\), first rewrite it as a reciprocal: \(\frac{1}{32^{\frac{4}{5}}}\).

Step 2: Simplify the Fractional Exponent

The fractional exponent \(\frac{4}{5}\) means taking the fifth root of the base (32) and then raising it to the fourth power. First, find the fifth root of 32: \(32^{\frac{1}{5}} = 2\), since \(2^5 = 32\).

Step 3: Raise to the Power of the Numerator

Now, take the result of the fifth root from Step 2 and raise it to the fourth power: \(2^4 = 16\).

Step 4: Take the Reciprocal

Finally, since we've simplified \(32^{\frac{4}{5}}\) to \(16\), remember to take the reciprocal due to the negative exponent. Therefore, \(32^{\frac{4}{5}} = 16\) implies \(32^{-\frac{4}{5}} = \frac{1}{16}\).

Key Concepts

Fractional Exponents

When you see a fractional exponent, like in the expression \(32^{-\frac{4}{5}}\), it might seem a bit tricky at first. But remember, it's just a shorthand way of doing two things in a row: finding a root and then raising to a power. For instance, \(32^{\frac{4}{5}}\) means you need to take the fifth root of 32 first and then raise the result to the fourth power.
This dual process is key to understanding why fractional exponents work the way they do. The denominator of the fraction (5, in this case) tells you which root to take, while the numerator (4 here) tells you the power you need to raise your result to after finding the root. This gives you a step-by-step path to follow when simplifying such expressions.

Reciprocal

A reciprocal is simply one divided by a number. When dealing with exponents, a negative exponent means you should take the reciprocal of the positive exponent's result. This means flipping everything upside down in fraction form.
For example, in the expression \(32^{-\frac{4}{5}}\), the negative sign in the exponent tells us we'll end up taking the reciprocal. So, once we find \(32^{\frac{4}{5}} = 16\), we need to find \(\frac{1}{16}\), giving us the final answer. This maneuver is what makes dealing with negative exponents a bit like flipping your math upside down!

Evaluating Expressions

Evaluating an expression means you calculate it to find a numerical value. In this case, evaluating \(32^{-\frac{4}{5}}\) happens step by step. First, consider the negative and fractional parts individually.
For the problem at hand, decompose \(32^{-\frac{4}{5}}\) into \(\frac{1}{32^{\frac{4}{5}}}\), based on the negative exponent rule. Then proceed to find \(32^{\frac{4}{5}}\) using the fractional exponent steps. This breakdown simplifies what seems complex at first. Once you handle each aspect separately, the final evaluation becomes straightforward.

Simplifying Expressions

Simplifying expressions with exponents is crucial for getting to the cleanest form of an answer. Start by addressing any fractional or negative exponents by breaking them down into more manageable parts.
In \(32^{-\frac{4}{5}}\), start by rewriting it to deal with the negative first as \(\frac{1}{32^{\frac{4}{5}}}\). Then focus on simplifying \(32^{\frac{4}{5}}\) through the series of root and power steps. Each move takes you closer to the simplified form, and in this case, simplifying eventually leads to \(\frac{1}{16}\). This process turns what seems complex into an understandable and solvable problem.