Problem 83 - College Algebra

Question

In Exercises \(77-84,\) evaluate each expression without using a calculator. $$32^{-4 / 5}$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Expression With Negative Exponent

The first step is to rewrite the expression using the rule \(a^{-m} = 1/a^m\). So, \(32^{-4 / 5}\) can be written as \(1/32^{4 / 5}\).

Step 2: Simplify the Expression with Fractional Exponent

Next, observe that the expression in the denominator is in the form \(a^{m/n}\), which can be written as the \(n\)th root of \(a^m\). Therefore, \(32^{4 / 5}\) can be written as the 5th root of \(32^4\). Now, 32 can be written as 2 to the power of 5, so the expression becomes the 5th root of \((2^5)^4\) which simplifies to the 5th root of \(2^{20}\). The 5th root of \(2^{20}\) is \(2^{20/5}\) = \(2^4\) = 16

Step 3: Write the Final Answer

Given that the original expression was the reciprocal of \(32^{4 / 5}\), take the reciprocal of 16. So, \(32^{-4 / 5}\) = \(1/16\).

Key Concepts

Negative Exponents

Negative exponents might seem intimidating at first, but once you understand the basic concept, they become much simpler to handle. When you see an expression like \(a^{-m}\), it tells you to take the reciprocal of \(a^m\). Essentially, any non-zero number raised to a negative exponent is equal to one divided by that number raised to the corresponding positive exponent. For example:

• \(a^{-3} = \frac{1}{a^3}\)
• \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)

In our exercise, \(32^{-4/5}\), the negative exponent means we take the reciprocal of \(32^{4/5}\). This is a critical first step in simplifying expressions with negative exponents.

Fractional Exponents

Fractional exponents are another way to express roots and powers combined. They are written in the form \(a^{m/n}\), where:

• \(m\) is the power to which the base \(a\) is raised
• \(n\) is the root we are taking

So, \(32^{4/5}\) means the 5th root of \(32\) raised to the 4th power. To simplify, notice that 32 is \(2^5\). When raised to the 4/5 power, it becomes the 5th root of \((2^5)^4\), which is \(2^4\) or 16.This approach of rewriting fractional exponents helps simplify calculations while keeping track of both roots and powers involved.

Roots and Powers

Understanding how roots and powers work together is key to simplifying expressions effectively. When we deal with roots, we are essentially asking what number, when raised to a certain power, gives us the original number. Meanwhile, powers indicate how many times we multiply the base by itself. For example, the 5th root of a number \(a\), denoted as \(\sqrt[5]{a}\), asks for a number which multiplied by itself 5 times results in \(a\).
In our example of \(32^{4/5}\), seeing the base 32 as \(2^5\) makes it easier to handle the root and power:

• The power 4 remains with the base after simplifying the root.
• It results in \(2^4\) since \(\sqrt[5]{2^{20}} = 2^4\).

These operations can initially seem complex, but breaking them down helps in understanding how roots and powers can be used interchangeably to simplify expressions.

Simplifying Expressions

Simplifying expressions is about rewriting them in a form that is easier to understand or calculate. The process involves using a combination of algebraic rules, such as dealing with exponents, roots, and simplifying fractions. In our example:

• We first use the negative exponent rule to find the reciprocal of \(32^{4/5}\).
• Then, change the fractional exponent into a root and power expression to make calculations easier.
• Finally, simplify the expression to its lowest terms.

For \(32^{-4/5}\):

• Convert the negative exponent to \(\frac{1}{32^{4/5}}\).
• Simplify \(32^{4/5}\) to \(2^4\).
• The final simplified expression is \(\frac{1}{16}\).

By systematically applying these simplification techniques, we ensure precision in calculations and clarity in understanding complex expressions.