Problem 45 - College Algebra with Modeling and Visualization

Question

Write the expression in radical notation. $$ y^{-1 / 5} $$

Step-by-Step Solution

Answer

The expression is \( \frac{1}{\sqrt[5]{y}} \).

Step 1: Understand the Problem

The problem requires us to convert the expression from an exponent form to a radical notation. The expression given is \( y^{-1/5} \).

Step 2: Recall Exponent Rules

We need to recall that a negative exponent \( a^{-n} \) can be expressed as \( \frac{1}{a^n} \). Therefore, \( y^{-1/5} = \frac{1}{y^{1/5}} \).

Step 3: Express Exponent as Radical

Now, express \( y^{1/5} \) in radical notation. Recall that \( a^{1/n} \) is equivalent to the nth root of \( a \). Thus, \( y^{1/5} \) becomes \( \sqrt[5]{y} \).

Step 4: Write Expression in Radical Notation

Combine the results from the previous steps, \( y^{-1/5} = \frac{1}{y^{1/5}} = \frac{1}{\sqrt[5]{y}} \). Hence, the expression in radical notation is \( \frac{1}{\sqrt[5]{y}} \).

Key Concepts

Negative Exponents

When dealing with exponents, a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. This means that for any non-zero number \( a \), \( a^{-n} \) can be rewritten as \( \frac{1}{a^n} \). This transformation is crucial to simplifying expressions with negative exponents. Here are a few key points about negative exponents:

• Negative exponents suggest that the base is on the incorrect side of the fraction line and need to be moved.
• Perform the conversion to make computations easier and to express them in a more familiar mathematical form.
• This concept applies for any number or variable, making it a versatile tool in algebra.

For example, in the case of \( y^{-1/5} \), its equivalent form using negative exponent rule is \( \frac{1}{y^{1/5}} \). Understanding how to handle negative exponents is essential in converting expressions to more useful forms.

Exponent Rules

Exponents have several important rules that facilitate the manipulation and simplification of algebraic expressions. Here are a few fundamental exponent rules:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \), provided \( a eq 0 \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Zero Exponent Rule: \( a^0 = 1 \), where \( aeq 0 \)

These rules enable the rewriting of expressions in different forms, allowing for simplification or conversion to radicals. Applying these rules to \( y^{-1/5} \) allows us to separate the negative exponent from the positive, illustrating a step to the radical transformation as \( \frac{1}{y^{1/5}} \). These foundational rules underpin much of algebraic manipulation and are vital for working efficiently with exponential expressions.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and nth roots. They are often presented using the radical symbol \( \sqrt{} \). Here are some important points about radical expressions:

• A radical expression like \( a^{1/n} \) represents the nth root of \( a \), written as \( \sqrt[n]{a} \).
• Radicals can be rewritten using exponents, with \( \sqrt[n]{a} = a^{1/n} \).
• Radical expressions are useful for representing fractional powers in a more perceptible form.

In the problem provided, the expression \( y^{1/5} \) is expressed as \( \sqrt[5]{y} \). Using radical notation signifies understanding the root aspect of the expression, further illustrating the relationship between exponents and radicals. Finally, combining these steps leads to \( y^{-1/5} = \frac{1}{\sqrt[5]{y}} \), presenting the expression in clear radical form.