Problem 63 - College Algebra

Question

Simplify the radical expressions in Exercises \(61-68\) $$\sqrt[3]{x^{4}}$$

Step-by-Step Solution

Answer

\(x^{4/3}\)

Step 1: Express the cubic root as an exponent

The cubic root of \(x^{4}\) can be expressed as \(x^{4/3}\). This is based on the principle that the nth root of a can be written as \(a^{1/n}\). Hence, \(\sqrt[3]{x^{4}}\) is equivalent to \(x^{4/3}\).

Step 2: Simplify the exponent

The term \(x^{4/3}\) is already in its most simplified form because the exponent \(4/3\) cannot be simplified further. Hence, the simplified form of the given radical expression is \(x^{4/3}\).

Key Concepts

Cubic Root

Understanding the concept of a cubic root is essential when dealing with radical expressions. A cubic root of a number is a value that, when multiplied by itself three times (or raised to the power of three), gives the original number. For instance, the cubic root of 8 is 2, symbolized as \(\sqrt[3]{8} = 2\), because \(2 \times 2 \times 2 = 8\).

When working with variables, the cubic root can also be applied. For a variable \(x\) raised to any power, the cubic root is represented as \(\sqrt[3]{x}\). In the exercise provided, you are asked to simplify \(\sqrt[3]{x^{4}}\), which involves finding a number that, when cubed, will result in \(x^{4}\). It's crucial to comprehend this process of 'undoing' the cube to grasp how to handle rational exponents, which you'll encounter next.

Exponent

An exponent tells us how many times a number or variable is used as a factor in a multiplication. It is written as a small number to the top right of the base number. For example, \(x^3\) means \(x\) is multiplied by itself three times: \(x \times x \times x\).

Exponents are not just whole numbers. They can be fractions, which leads us to an interesting part about rational exponents. For instance, an exponent of \(\frac{1}{2}\) signifies a square root, and \(\frac{1}{3}\) indicates a cubic root. In the textbook problem, \(x^{4/3}\) represents the original expression \(\sqrt[3]{x^{4}}\) as an exponent with a numerator and denominator to illustrate the root and power relationship effectively.

Radical Exponentiation

Radical exponentiation is a method to express roots, such as square roots or cubic roots, with exponents. This technique uses rational (or fractional) exponents, as we've seen with the cubic root being written as \(x^{1/3}\). This is useful because it allows us to apply the rules of exponents to simplify radical expressions.

When you encounter a radical with an exponent, like \(\sqrt[3]{x^{4}}\), you transform it into the expression \(x^{4/3}\). The conversion simplifies the operation, enabling further algebraic manipulation using familiar exponent rules such as the product rule, quotient rule, and power rule. This strategy streamlines solving equations and can be invaluable in higher-level mathematics, including calculus. Simplifying \(\sqrt[3]{x^{4}}\) to \(x^{4/3}\), as in our example, takes advantage of radical exponentiation and paves the way for more advanced algebraic processes.