Problem 16 - Intermediate Algebra

Question

Use radical notation to write each expression. Simplify if possible. $$ (-32)^{1 / 5} $$

Step-by-Step Solution

Answer

The simplified form is -2.

Step 1: Understand Radical Notation

Radical notation involves writing expressions with roots using the radical symbol. The expression \((-32)^{1/5}\) is equivalent to saying "the fifth root of -32."

Step 2: Convert to Radical Notation

Write the expression \((-32)^{1/5}\) using radical notation. The expression becomes \(\sqrt[5]{-32}\), which reads as "the fifth root of -32."

Step 3: Calculate the Fifth Root

Determine the number that, when raised to the fifth power, results in -32. Since \((-2)^5 = -32\), the fifth root of \(-32\) is \(-2\).

Step 4: Simplification

Since the fifth root of -32 is -2, the simplified form of the expression in radical notation is -2.

Key Concepts

Fifth Roots

The concept of fifth roots is a fascinating part of mathematics that involves finding a number, which, when multiplied by itself five times, returns the original value you're interested in. Specifically, for any given number \(x\), the fifth root, often expressed as \(\sqrt[5]{x}\), refers to a number \(a\) such that \(a^5 = x\).
For a practical example, let's consider \((-32)\). When you take the fifth root of \(-32\), you are seeking a number that, when raised to the power of five, equals \(-32\). In this case, that number is \(-2\), because:

• \((-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\).

Remember, taking roots of negative numbers is only possible with odd roots like the fifth root, as they result in negative outputs.

Expression Simplification

Simplifying expressions is a crucial skill in mathematics that involves reducing an expression to its simplest form. This often means finding a number or expression that is easier to work with or more understandable without changing the value of the original expression.
In the case of our example \((-32)^{1/5}\), you're simplifying the expression by calculating what \((-32)\) results in when expressed under the fifth root, which is

• First, express the power in radical notation: \((-32)^{1/5} = \sqrt[5]{-32}\).
• Then calculate the fifth root: \(\sqrt[5]{-32} = -2\).

Hence, the simplification process converts a potentially complicated expression into a straightforward numerical result, \(-2\), that is much easier to understand and work with.

Exponentiation Concepts

Exponentiation is a key concept in algebra that involves repeated multiplication of a base number. When dealing with expressions like \((-32)^{1/5}\), you are indeed leveraging the fractional exponent's relationship to roots.
In exponential terms, \(x^{1/n}\) denotes the \(n\)th root of \(x\). Therefore, the expression \((-32)^{1/5}\) can be reinterpreted in radical terms as the fifth root, \(\sqrt[5]{-32}\). This concept shows the powerful relationship between roots and exponents, where a fractional exponent is essentially an indicator of a root.

• Fractional exponents connect directly to roots, allowing for flexible expression transformations.
• This relationship simplifies complex problems, making exponential expressions manageable.

Understanding these connections makes it easier to move between exponential and radical forms, enhancing problem-solving efficiency in mathematics.