Problem 524 - Elementary Algebra

Question

In the following exercises, write as a radical expression. (a) \(x^{\frac{1}{2}}\) (b) \(y^{\frac{1}{3}}\) (c) \(z^{\frac{1}{4}}\)

Step-by-Step Solution

Answer

(a) \(\sqrt{x}\), (b) \(\sqrt[3]{y}\), (c) \(\sqrt[4]{z}\)

Step 1: Understanding the problem

The task is to convert three expressions with fractional exponents into radical expressions. Each fractional exponent represents a root.

Step 2: Recall the rule for fractional exponents

Remember that any expression of the form \(a^{\frac{1}{n}}\) can be written as the n-th root of \(a\). Specifically, \(a^{\frac{1}{2}} = \sqrt{a}\), \(a^{\frac{1}{3}} = \sqrt[3]{a}\), and \(a^{\frac{1}{4}} = \sqrt[4]{a}\).

Step 3: Convert each given expression

We apply the rule from Step 2 to each expression: (a) \(x^{\frac{1}{2}} = \sqrt{x}\)(b) \(y^{\frac{1}{3}} = \sqrt[3]{y}\)(c) \(z^{\frac{1}{4}} = \sqrt[4]{z}\)

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, are a way to express roots using exponents. Instead of writing the square root or cube root using the radical symbol, we use a fraction as the exponent. This is a crucial concept in algebra as it simplifies many expressions and makes it easier to perform algebraic operations.
For example, the expression \(x^{\frac{1}{2}}\) represents the square root of \(x\). The denominator of the fraction indicates the root. So, \(y^{\frac{1}{3}}\) means the cube root of \(y\), and \(z^{\frac{1}{4}}\) stands for the fourth root of \(z\).
Here’s a quick summary:

• \[x^{\frac{1}{n}} = \sqrt[n]{x}\]
• The denominator of the fractional exponent (n) indicates the degree of the root.

This method allows us to seamlessly transition between exponential and radical expressions, enhancing our ability to work with complex equations and simplifying the overall algebraic process.

Radical Notation

Radical notation is the traditional way to denote roots and it involves the use of the radical symbol (√). The number placed under the radical sign is called the radicand, and the number outside the radical represents the index of the root.
For instance:

• \[\sqrt{x}\]: This is the square root of \(x\) and can be written as \(x^{\frac{1}{2}}\).
• \[\sqrt[3]{y}\]: This is the cube root of \(y\) and can be written as \(y^{\frac{1}{3}}\).
• \[\sqrt[4]{z}\]: This is the fourth root of \(z\) and can be written as \(z^{\frac{1}{4}}\).

Understanding and converting between radical notation and fractional exponents is essential in solving algebraic equations since it provides two different perspectives for tackling the same problem. By practicing these conversions, students can gain a deeper understanding of how these mathematical operations are interconnected.

Equivalent Expressions

Equivalent expressions are different mathematical expressions that simplify to the same value. In the context of radical expressions and fractional exponents, this means that any root in radical form can be represented by a fractional exponent and vice versa.
This equivalence allows for flexibility in calculations and simplification of algebraic expressions.
For example:

• \[x^{\frac{1}{2}} = \sqrt{x}\]
• \[y^{\frac{1}{3}} = \sqrt[3]{y}\]
• \[z^{\frac{1}{4}} = \sqrt[4]{z}\]

Recognizing these equivalences can make solving algebraic problems easier. When working with polynomials, roots, or complex numbers, utilizing equivalent expressions can help in factoring, simplifying, or solving equations more efficiently. By mastering both the radical notation and the use of fractional exponents, students can approach problems from multiple angles to find the most straightforward solution.