Problem 32 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The value is \( \frac{1}{8} \).

Step 1: Rewrite the Fractional Exponent

The expression given is \( \left(\frac{1}{32}\right)^{3/5} \). We can use the property of exponents, which states that \( a^{m/n} = (a^m)^{1/n} = (a^{1/n})^m \), to rewrite the expression. This means we will separate the powers as: \( \left(\frac{1}{32}\right)^{3/5} = \left(\left(\frac{1}{32}\right)^{1/5}\right)^3 \).

Step 2: Find the Fifth Root

Now, we need to find the fifth root of \( \frac{1}{32} \). Recall that the fifth root of a fraction is the same as taking the fifth root of both the numerator and the denominator separately: \( (\frac{1}{32})^{1/5} = \frac{1^{1/5}}{32^{1/5}} \). Since \( 1^{1/5} = 1 \), the expression simplifies to \( \frac{1}{32^{1/5}} \).

Step 3: Calculate the Fifth Root of 32

Find the fifth root of 32. Since \( 32 = 2^5 \), it follows that \( 32^{1/5} = (2^5)^{1/5} = 2 \). Thus, \( \frac{1}{32^{1/5}} = \frac{1}{2} \).

Step 4: Apply the Remaining Power

Finally, apply the cube: \( \left(\frac{1}{2}\right)^3 \). This means multiplying \( \frac{1}{2} \) by itself three times: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \).

Key Concepts

Properties of Exponents

When dealing with expressions involving exponents, understanding their properties is crucial. Exponents represent how many times a number, called the base, is multiplied by itself. Fractional exponents reveal a special situation where the exponent can be a fraction, like \( \frac{3}{5} \). This kind of exponent has a dual role as both a power and a root.

For any number \( a \) raised to the fraction \( \frac{m}{n} \), you can think of it as \( a^{m/n} = (a^m)^{1/n} = (a^{1/n})^m \). This means:

• \( a^{1/n} \) signifies taking the \( n \)-th root of \( a \).
• \( a^m \) means you multiply \( a \) by itself \( m \) times.

The property that \( (a^{1/n})^m = a^{m/n} \) allows us to break down complex fractional exponents into simpler steps, as seen in the solution.

Simplifying Radicals

Radicals involve roots, such as square roots, cube roots, and beyond. Simplifying radicals makes them easier to work with in expressions. In our exercise, we tackled the fifth root of the fraction \( \frac{1}{32} \).

To simplify, we take each part separately:

• The fifth root of the numerator \( 1 \) is straightforward because any root of \( 1 \) is still \( 1 \).
• The fifth root of \( 32 \) uses its prime factorization: \( 32 = 2^5 \). Therefore, \( 32^{1/5} = (2^5)^{1/5} = 2 \).

Thus, the fifth root of \( \frac{1}{32} \) simplifies to \( \frac{1}{2} \). Simplifying radicals involves recognizing the type of root and applying that understanding to break down the expression.

Basic Arithmetic Operations

Arithmetic operations are foundational in solving any expression. They include addition, subtraction, multiplication, and division. These operations often come into play after breaking down complex expressions, like those with exponents.

In the last step of the example, after simplifying the radical, we needed to cube \( \frac{1}{2} \), which involved basic multiplication:

• Multiply \( \frac{1}{2} \) by itself three times: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \), then \( \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \).

This demonstrates how understanding simple arithmetic ensures the correct simplification of expressions with fractional exponents and radicals, bringing us to the complete solution.