Problem 65 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \frac{1}{5^{\frac{1}{2}}} $$

Step-by-Step Solution

Answer

The expression is \( \frac{1}{\sqrt{5}} \) in radical form.

Step 1: Understand the Given Expression

The expression given is \( \frac{1}{5^{\frac{1}{2}}} \). This expression involves a power with a fractional exponent.

Step 2: Understand Radical Terminology

A fractional exponent like \( \frac{1}{2} \) can be expressed as a radical. Specifically, \( a^{\frac{1}{n}} \) is equivalent to \( \sqrt[n]{a} \). Here, \( n \) is the denominator of the fraction.

Step 3: Convert to Radical Form

The term \( 5^{\frac{1}{2}} \) can be expressed as \( \sqrt{5} \), because \( \frac{1}{2} \) indicates the square root.

Step 4: Write the Expression as a Radical

Replace \( 5^{\frac{1}{2}} \) with \( \sqrt{5} \) in the original expression. This gives: \( \frac{1}{\sqrt{5}} \).

Step 5: Simplify the Expression (if necessary)

The expression \( \frac{1}{\sqrt{5}} \) is already in its simplest radical form. For simplification into a rationalized form, multiply numerator and denominator by \( \sqrt{5} \): \( \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \). However, in the context of expressing as a radical form, \( \frac{1}{\sqrt{5}} \) is acceptable.

Key Concepts

Radical Expressions

A radical expression is an expression that contains a radical symbol (√), which represents the root of a number. In mathematics, radicals provide a different way of expressing powers, especially when dealing with fractional exponents.

Understanding how to convert between radical expressions and expressions with fractional exponents is essential. For instance, an expression such as \( a^{\frac{1}{n}} \) translates to the nth root of \( a \), denoted as \( \sqrt[n]{a} \). The denominator of the fractional exponent determines the type of root: a square root if the denominator is 2, a cube root if it’s 3, and so forth.

When encountered with fractional exponents, converting them into radical form can often make it easier to visualize and work with the numbers. This conversion is especially useful in simplifying terms or solving equations that involve roots.

• Square root: \( a^{\frac{1}{2}} = \sqrt{a} \)
• Cube root: \( a^{\frac{1}{3}} = \sqrt[3]{a} \)
• n-th root: \( a^{\frac{1}{n}} = \sqrt[n]{a} \)

By recognizing these equivalencies, it becomes straightforward to switch between exponents and radicals to suit the needs of your problem.

Simplifying Radicals

Simplifying radicals involves reducing the radical expression to its simplest form, making the numbers inside the radical sign as small as possible. The goal is to present the expression in an easily understandable form without altering its value.

To simplify a radical, you need to factor the number inside the radical to find perfect squares (or cubes, etc., depending on the root). For example:

• \( \sqrt{18} \) can be simplified by recognizing \( 18 = 9 \times 2 \). Since 9 is a perfect square, \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \).

In the exercise presented, \( \frac{1}{\sqrt{5}} \) is already simplified in radical expression form. However, sometimes further simplification can involve rationalizing the denominator, which will be discussed in the next section.

Always aim to simplify radicals as this maintains the integrity of mathematical operations and makes the expressions more manageable.

Rationalization

Rationalization is the process of eliminating radicals from the denominator of a fraction. This is typically preferred in mathematics because fractions are easier to interpret when they contain only rational numbers.

In the provided exercise, we have the expression \( \frac{1}{\sqrt{5}} \). The denominator has a square root, which makes it an irrational number. To rationalize it, follow these steps:

• Multiply both the numerator and the denominator by the square root that is in the denominator, \( \sqrt{5} \).
• This multiplication gives \( \frac{1 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{\sqrt{5}}{5} \).

This process changes the square root in the denominator into a whole number, thanks to the property that \( \sqrt{a} \times \sqrt{a} = a \).

The expression \( \frac{\sqrt{5}}{5} \) is considered rationalized because the denominator is now a rational number. Rationalization makes further calculations more straightforward and is especially useful when adding, subtracting, or comparing radical expressions.