Problem 6 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ 32^{1 / 5} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Expression

The given expression is a fractional exponent: \(32^{1/5}\). This represents the fifth root of 32, as fractional exponents denote roots. \(a^{1/n} = \sqrt[n]{a}\).

Step 2: Finding the Fifth Root

We need to find a number that, when raised to the power of 5, equals 32. Consider small integers to find this number. \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\). Thus, \(2^5 = 32\), so \(32^{1/5} = 2\).

Step 3: Evaluating the Expression

Given that raising 2 to the fifth power results in 32, the fifth root of 32 is 2. Therefore, \(32^{1/5}\) simplifies directly to 2.

Key Concepts

Roots and Radicals

Roots and radicals are foundational concepts in algebra. Root operations involve finding a number that, when raised to a specified power, returns the initial number. For instance, in the case of a square root, we find a number that multiplied by itself returns the original number. Similarly, a cubic root finds a number that when used thrice in multiplication returns the original number.
Fractional exponents provide a concise way to represent radicals. Instead of using the square root symbol, we can use an exponent, such as in the case of \(a^{1/5}\). This denotes the fifth root of \(a\), as demonstrated in the exercise with the expression \(32^{1/5}\).

• The denominator in a fractional exponent indicates the root.
• The numerator, if greater than 1, indicates the power to which the root result is raised.

Understanding roots and radicals helps simplify and solve complex algebraic expressions. It also reveals the flexibility and simplicity that fractional exponents introduce in expressing roots.

Exponential Expressions Evaluation

Evaluating exponential expressions involves understanding the relationship between the base and its exponent. With fractional exponents, comprehension of both the power and the root operations is necessary.
When you encounter an expression like \(32^{1/5}\), you translate it to its root form, \(\sqrt[5]{32}\). Here, you seek to find a base number that, multiplied by itself five times, equals 32.

• Convert the fractional exponent into a root expression.
• Identify the base number by testing small integers efficiently.
• Remember that finding a root reverses the process of exponentiation.

Through practice, evaluating these expressions becomes intuitive. The evaluated result connects the initial fractional exponent with its simple, whole-number outcome. Understanding how to efficiently evaluate these expressions is invaluable for algebra and higher-level mathematics studies.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging and simplifying expressions to solve problems or understand equations better. This skill is crucial in working through expressions with fractional exponents, like the one seen in the exercise.
When you are faced with an expression such as \(32^{1/5}\), you're engaging both numeric and algebraic manipulation. Here, you transcribe the expression into its root form and simplify it to get a clear, concise result.

• Rewrite expressions using properties of exponents and roots for clarity.
• Perform basic operations like multiplication and division to test potential solutions.
• Use logical reasoning to verify your results and confirm accuracy.

These strategies are not only practical for solving equations but also develop deeper insights into mathematical properties and structures. Mastering algebraic manipulation enriches your problem-solving toolkit and enhances analytical thinking.