Problem 10 - Elementary and Intermediate Algebra

Question

Assume for all exercises that even roots are of non- negative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. $$ y^{1 / 5} $$

Step-by-Step Solution

Answer

The equivalent expression is \( \sqrt[5]{y} \).

Step 1: Identify the Exponent

Recognize that the given expression is in exponential form: \( y^{1/5} \).

Step 2: Convert to Radical Notation

Convert the expression from exponential form to radical form using the property \( a^{1/n} = \sqrt[n]{a} \). Thus, \( y^{1/5} = \sqrt[5]{y} \).

Step 3: Simplify if Possible

Check if the radical can be simplified further. Since \( \sqrt[5]{y} \) is already in its simplest form, no further simplification is possible.

Key Concepts

Exponent

An exponent is a number that tells us how many times to multiply a base number by itself. In the expression,\( y^{1/5} \), the base is \( y \) and the exponent is \( \frac{1}{5}\). This means we are taking the fifth root of \( y \).

Understanding exponents is crucial because they allow us to express large numbers concisely and work with roots efficiently. Here, the exponent \( \frac{1}{5} \) is a fraction. Fractional exponents denote roots. Specifically, the denominator of the fraction represents the root.

So, \( y^{1/5} \) means the fifth root of \( y \).

Simplification

Simplification involves reducing an expression to its most basic form. This step helps make mathematical problems easier to manage. In our case, simplifying an expression containing radicals or exponents helps us see the problem more clearly.

To simplify \( y^{1/5} \), we first convert it to radical form. Once we recognize that \( y^{1/5} \) is already in its simplest form as \( \sqrt[5]{y} \), no additional steps are needed.

Remember:

• Look for common factors
• Check if the numbers can be broken down further
• Understand that some expressions, like \( \sqrt[5]{y} \), are already as simple as they can get

Radicals

Radicals are symbols used to represent roots of numbers. The radical sign (\( \sqrt{} \)) is used to denote square roots, while higher order roots include an index, like \( \sqrt[5]{y} \) for the fifth root of \( y \).

Converting from exponential to radical notation helps in visualizing the problem. For instance, \( y^{1/5} \) converts to \( \sqrt[5]{y} \).

Key points to remember about radicals:

• The index outside the radical sign indicates the root's degree (e.g., 5 in \( \sqrt[5]{y} \)).
• The expression under the radical sign is called the radicand (e.g., \( y \)).
• Simplifying radicals may involve factoring the radicand or recognizing patterns.

Understanding radicals is essential for working with roots in algebra and beyond.