Problem 16 - Elementary Algebra

Question

Express in radical form. \(x 5 / 6\)

Step-by-Step Solution

Answer

The radical form of \(x^{\frac{5}{6}}\) is \(\sqrt[6]{x^5}\).

Step 1: Understand the expression

The expression provided is \(x^{\frac{5}{6}}\). This type of expression is known as an exponent with a fractional power, and it can be converted into a radical form.

Step 2: Apply the rule for converting fractional exponents to radicals

According to the rule, an expression \(x^{\frac{m}{n}}\) can be rewritten in radical form as \(\sqrt[n]{x^m}\). This means the denominator of the fraction is the index of the radical, and the numerator is the power inside the radical.

Step 3: Rewrite the given expression

Using the rule from Step 2, rewrite \(x^{\frac{5}{6}}\) into its radical form. The denominator is 6, which becomes the index of the radical, and the numerator is 5, which remains the power of \(x\) inside the radical. This gives us \( \sqrt[6]{x^5} \).

Key Concepts

Fractional Exponents

When dealing with fractional exponents, it's important to remember that they are a way to express roots and powers in a concise form. Here’s how it works:

Fractional exponents are written as \( x^{\frac{m}{n}} \), where \( m \) and \( n \) are integers. The numerator \( m \) signifies the power of the base \( x \), while the denominator \( n \) represents the root or the degree of the radical.

Why use fractional exponents? They allow for a unified way of expressing both powers and roots. This is helpful in simplification and solving equations. By changing between radical and exponential forms, we can make calculations easier. For example:

• \( x^{\frac{1}{2}} \) is the same as \( \sqrt{x} \).
• \( x^{\frac{3}{4}} \) equals \( \sqrt[4]{x^3} \).

These examples show how fractional exponents provide a different yet powerful way of understanding roots and powers.

Exponents

Exponents are a fundamental tool in algebra that describe how many times a number or a variable is multiplied by itself. The expression \( x^n \) indicates that \( x \) is the base, and \( n \) is the exponent.

Understanding exponents is crucial for mastering algebra. Exponents can be:

• **Positive integers**, such as \( x^3 \) (which means \( x \times x \times x \)).
• **Negative integers**, such as \( x^{-3} \) (which is \( \frac{1}{x^3} \)).
• **Zero**, where \( x^0 = 1 \) for all non-zero \( x \).
• **Fractions**, as in \( x^{\frac{m}{n}} \), introducing roots and radicals.

Successfully manipulating expressions with exponents requires knowledge of several rules, such as the power rule, product rule, and quotient rule. This enables simplification and transformation of expressions, which is foundational in solving algebraic problems.

Radical Form

Radical form involves expressions with roots, such as square roots, cube roots, or higher. These are often represented as \( \sqrt[n]{x} \), where \( n \) is the index of the root.

The radical sign "\( \sqrt{} \)" represents the root. It's important to match the structure of roots with corresponding fractional exponents:

• The **square root** of \( x \) is written as \( \sqrt{x} \) or equivalently \( x^{\frac{1}{2}} \).
• The **cube root** of \( x \) is \( \sqrt[3]{x} \) or \( x^{\frac{1}{3}} \).
• For any general root \( \sqrt[n]{x} \), the corresponding fractional exponent is \( x^{\frac{1}{n}} \).

Radicals offer an alternative way to represent the power of an expression that might be less obvious in its exponential form. This conversion between formats—especially between radicals and fractional exponents—is particularly useful in both solving and simplifying expressions.