Problem 16 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$32^{1 / 5}$$

Step-by-Step Solution

Answer

The value of \(32^{1/5}\) is 2.

Step 1: Understanding the Problem

We want to evaluate the expression \(32^{1/5}\). This notation means we are looking for the fifth root of 32, which is the number that, when multiplied by itself five times, yields 32.

Step 2: Break Down into Prime Factors

Start by breaking down 32 into its prime factors. Note that 32 is a power of 2, so \(32 = 2 imes 2 imes 2 imes 2 imes 2 = 2^5\).

Step 3: Apply the Fractional Exponent

Since \(32 = 2^5\), we can rewrite \(32^{1/5}\) using the property of exponents: \( (2^5)^{1/5} = 2^{(5 imes 1/5)} = 2^1\).

Step 4: Simplify the Expression

Simplify \(2^1\) to get 2. This step confirms that the original expression \(32^{1/5}\) simplifies to 2.

Key Concepts

Fifth Root

The fifth root of a number is the value that, when multiplied by itself five times, equals that number.
Finding the fifth root is like asking, "What number do I multiply by itself five times to get back to the original number?"
For the number 32, the fifth root is 2, because \(2 \times 2 \times 2 \times 2 \times 2 = 32\).
Fifth roots are simply an extension of the more familiar square and cube roots. You can also represent the fifth root using fractional exponents, where the exponent is \(1/5\).
This idea brings us to the concept of fractional exponents, which we will explore next.

Prime Factorization

Prime factorization is the process of breaking down a number into its most basic building blocks — the prime numbers.
A prime number is a number greater than 1 that has no other divisors except for 1 and itself.
The prime factorization of 32 is an important step in simplifying expressions like \(32^{1/5}\).

• Start with the number 32.
• Divide by the smallest prime number (2) and continue dividing until you cannot go further.

You will find that \(32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\).
This representation is useful when applying rules of exponents, as it transforms complex numbers into easily manageably forms.

Fractional Exponents

Fractional exponents are a way of expressing roots using exponents.
They allow us to represent roots and powers as a single operation. For example, the fifth root of a number can be written as an exponent of \(1/5\).

• This representation stems from the basic property of exponents: \( (a^m)^n = a^{m \times n} \).
• When you see \(32^{1/5}\), you can think of it as \((32)^1\) raised to the power of \(1/5\).

The benefit of this approach is that it simplifies complex operations into straightforward calculations.
By converting roots into fractional exponents, you can easily handle more complex mathematical equations and calculations.