Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Use rational exponents to reduce the index of the radical.\(\sqrt[9]{x^{3}}\)

Step-by-Step Solution

Answer

The simplified form of given expression \(\sqrt[9]{x^{3}}\) with reduced index is \(x^{1/3}\).

Step 1: Write the Root as an Exponent

The general rule is any radical \(\sqrt[n]{a^m}\) can be rewritten in exponent form as \(a^{m/n}\). So, according to rule, \(\sqrt[9]{x^{3}}\) can be displayed as \(x^{3/9}\).

Step 2: Simplify the Exponent

The exponent \(3/9\) can be simplified into its reduced form by dividing both the numerator and the denominator by their greatest common divisor, which is 3. The reduced exponent will be \(1/3\). Hence, \(x^{3/9}\) simplifies to \(x^{1/3}\).

Step 3: Check the Result

The resulting value \(x^{1/3}\) matches with the format for writing exponents in reduced form, thereby verifying the correctness of the solution.

Key Concepts

Simplifying Exponents

Simplifying exponents involves taking a fraction like \(\frac{3}{9}\) and reducing it to its lowest terms. This makes calculations simpler and clearer. Begin by finding the greatest common divisor (GCD) of the numerator and the denominator. For \(3\) and \(9\), the GCD is \(3\). Divide both the numerator and the denominator by this number: \(\frac{3}{9} \div \frac{3}{3} = \frac{1}{3}\).
This fraction now represents the exponent in its simplest form. Simple fractions lead to easier calculations and frequently appear in various algebraic expressions.
It's important to simplify these exponents to obtain accurate and manageable equations that are friendly to work with.

Radical Expressions

Radical expressions are expressions that include a root, such as a square root \(\sqrt{}\) or a cube root \(\sqrt[3]{}\). They are a standard part of algebra and appear in many mathematical problems. For example, \(\sqrt[9]{x^3}\) is a radical expression with a degree of \(9\), indicating the ninth root.
This kind of expression can be complex to handle directly. That's where converting them to exponential form is beneficial. By using the relationship \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\), complicated roots can be expressed as exponents, which are often simpler to manipulate.
This conversion offers a uniform method to approach radicals, making solutions more straightforward and calculations faster.

Fractional Exponents

Fractional exponents are another way to express roots using exponents. They are written in the form \(a^{m/n}\), where the denominator represents the root and the numerator the power.
For instance, in \(x^{1/3}\), \(1/3\) indicates the cube root of \(x\). This representation is quite useful, as exponent rules apply, making it easier to simplify and manipulate expressions compared to using radical notation.

• Fraction \(\frac{1}{2}\) signifies a square root.
• Fraction \(\frac{1}{3}\) signifies a cube root.
• If the numerator was \(2\), like \(x^{2/3}\), it means squaring \(x\) first, then taking the cube root.

These expressions allow for a smoother integration into algebraic operations, making it easier to solve equations and work with exponential forms.