Problem 69 - College Algebra Essentials

Question

Simplify the radical expressions in Exercises \(67-74\) if possible. $$\sqrt[3]{x^{4}}$$

Step-by-Step Solution

Answer

The simplified version of the expression \(\sqrt[3]{x^{4}}\) is \(x*\sqrt[3]{x}\)

Step 1: Apply the definition of cube root

Rewrite the expression \(\sqrt[3]{x^{4}}\) as \(x^{4/3}\). This is due to the rule of indices that \(\sqrt[n]{a^m}=a^{m/n}\), this principle can be used to convert root form into fractional exponent.

Step 2: Simplify the fractional exponent.

The fraction \(4/3\) can be rewritten as \(1 + 1/3\) or \(1.33\). This is the same as saying \(x^1 * x^{1/3}\). Split \(x^{4/3}\) into two parts: \(x^1\) and \(x^{1/3}\)

Step 3: Write in radical form

Convert \(x^{1/3}\) back to its root form, and the expression can be rewritten as \(x*\sqrt[3]{x}\)

Key Concepts

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8, denoted as \( \sqrt[3]{8} \), is 2 because \( 2 \times 2 \times 2 = 8 \). When simplifying radical expressions with cube roots, such as \( \sqrt[3]{x^{4}} \), we're essentially looking for a number that, when cubed, gives us \( x^{4} \). This operation can expose students to the concept of fractional exponents, as the rules of indices allow us to represent cube roots in this manner.

In practice, cube roots can be challenging when the radicand (the number inside the radical) is not a perfect cube. In these cases, we must use properties of exponents to simplify the expression as much as possible. Understanding how to handle cube roots can be invaluable for solving various types of algebraic problems.

Fractional Exponents

Fractional exponents represent roots and power in one expression. The numerator indicates the power, while the denominator indicates the root. For example, the expression \( x^{1/3} \) signifies the cube root of \( x \) or \( \sqrt[3]{x} \).

Simplifying fractional exponents involves breaking them down into more manageable parts, usually through basic algebraic rules. If we have an expression like \( x^{4/3} \), we recognize that this can be split into \( x^{1} \) times \( x^{1/3} \). This division of the exponent allows us to deal with integer exponents and roots separately, simplifying our path to a solution in a step-by-step, digestible approach.

Indices

Indices, also known as exponents or powers, tell us how many times a number, known as the base, is multiplied by itself. The expression \( x^{4} \) means \( x \) is multiplied by itself four times. A basic understanding of indices is essential in algebra, as it helps us manipulate expressions involving powers.

One of the core rules when dealing with indices involves the manipulation of root expressions by converting them into exponents. The rule \( \sqrt[n]{a^m} = a^{m/n} \) is a pivotal concept, enabling us to translate between radical form and exponential form. This conversion is a fundamental step in simplifying radical expressions, making indices a cornerstone concept in algebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. Simplifying these expressions is often about combining like terms and applying algebraic rules to reduce the expression to its simplest form.

When dealing with radical expressions, such as cube roots, we apply algebraic rules to simplify them. These rules can include the distribution of exponents over multiplication, understanding the properties of roots and powers, and more. Mastery of how to handle algebraic expressions can enable students to tackle a wide range of problems and is a critical skill in both educational and real-world applications.