Problem 32 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{1}{32}\right)^{3 / 5} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{8} \).

Step 1: Understand the Expression

The expression \( \left( \frac{1}{32} \right)^{3/5} \) is a fractional exponent and involves both power and root. The base here is \( \frac{1}{32} \), and the exponent is \( \frac{3}{5} \). The expression combines both a cube and a fifth root.

Step 2: Simplify the Base

Recognize that \( \frac{1}{32} \) can be rewritten as the power of a number. Since \( 32 = 2^5 \), we have \( \frac{1}{32} = 2^{-5} \). Therefore, the expression becomes \( (2^{-5})^{3/5} \).

Step 3: Apply the Power of a Power Property

Use the property of exponents which states that \((a^m)^n = a^{m \cdot n}\). Apply this to our expression: \( (2^{-5})^{3/5} = 2^{-5 \cdot \frac{3}{5}} = 2^{-3} \).

Step 4: Calculate \(2^{-3}\)

Now calculate \( 2^{-3} \). By definition, \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \). Thus, the expression evaluates to \( \frac{1}{8} \).

Key Concepts

Fractional Exponents

Fractional exponents are expressions where an exponent is presented as a fraction, such as \( a^{m/n} \). This type of exponent involves two mathematical operations: a power and a root. It might seem complicated at first, but it's simpler when broken down.
For example, \( a^{m/n} \) can be interpreted as the \( n \)-th root of \( a \) raised to the \( m \)-th power, meaning:

• The number \( a \) is first taken to the \( m \)-th power.
• Then, the result is taken to the \( n \)-th root.

Alternatively, you could take the \( n \)-th root of \( a \) first and then raise it to the \( m \)-th power. Both interpretations yield the same result.
With fractional exponents, the denominator indicates the root, while the numerator represents the power, which helps in simplifying expressions systematically.

Exponent Properties

Understanding the properties of exponents is crucial for dealing with complex expressions, such as those with fractional exponents and negative bases. Knowing and applying these properties can make solving exponent problems easier.
Key properties include:

• **Product of Powers Property**: \( a^m \times a^n = a^{m+n} \).
• **Quotient of Powers Property**: \( \frac{a^m}{a^n} = a^{m-n} \) provided \( a eq 0 \).
• **Power of a Power Property**: \( (a^m)^n = a^{m\cdot n} \), which is especially useful in simplifying expressions like the exercise given.
• **Power of a Product Property**: \((ab)^m = a^m \cdot b^m\).
• **Negative Exponent Property**: \( a^{-m} = \frac{1}{a^m} \), interpreting negative exponents as reciprocals helps in simplification.

By familiarizing yourself with these properties, dealing with exponent-based expressions becomes a more straightforward task.

Simplifying Expressions

Simplifying expressions, especially those involving exponents, is about rewriting them in their simplest or most convenient form. Here’s a straightforward approach:

• Identify any bases that can be rewritten, as seen when converting \( \frac{1}{32} \) into \( 2^{-5} \).
• Apply relevant exponent properties like the Power of a Power property to simplify, as done with \( (2^{-5})^{3/5} \).
• Calculate further to reach a conclusive result. For instance, transforming \( 2^{-3} \) into \( \frac{1}{2^3} \) for simplicity.

The goal is to reduce each expression step-by-step to understand it and reach the smallest or most precise form, such as turning complex fractional exponent expressions into simple fractions or integers. Simplifying is the art that combines logic, properties, and computation for clarity and ease.