Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Use rational exponents to reduce the index of the radical.\(\sqrt[4]{\left(3 x^{2}\right)^{4}}\)

Step-by-Step Solution

Answer

The simplified expression of \(\sqrt[4]{(3x^{2})^{4}}\) is \(3x^2\).

Step 1: Rewrite the radical as a rational exponent

The expression can be rewritten using the rule \(\sqrt[n]{a^{m}} = a^{\frac{m}{n}}\). In this case, \(a = (3x^2)\), \(m = 4\), and \(n = 4\), so the expression can be rewritten as \((3x^2)^{\frac{4}{4}}\).

Step 2: Simplify the exponent

The fraction \(\frac{4}{4}\) simplifies to 1. So the expression becomes \((3x^2)^{1}\).

Step 3: Simplify the expression

Since any number/quantity raised to the power of 1 gives the number/quantity itself, the expression simplifies to \(3x^2\).

Key Concepts

Radical Reduction

Understanding radical reduction is crucial for simplifying expressions involving roots. This process involves transforming a radical expression into a simpler or more convenient form. A key method for radical reduction is using rational exponents. Rational exponents are expressions of the form \( a^{\frac{m}{n}} \), where \( n \) is the index of the radical and \( m \) is the exponent inside the radical. For instance, \( \sqrt[4]{(3x^{2})^{4}} \) can be represented as \( (3x^{2})^{\frac{4}{4}} \). By rewriting radicals as exponents, we enable the application of exponent rules to simplify the expression.

Let's not overlook the importance of understanding the index of a radical, which refers to the root degree (i.e., for a square root, the index is 2; for a cube root, the index is 3). When we encounter an expression like \( \sqrt[n]{a^{m}} \), our job is to reduce the index—this means either finding an equivalent expression with a smaller index or eliminating the radical altogether if the index and the exponent are the same, as seen in the exercise example.

Exponent Simplification

Exponent simplification removes complexities from expressions with powers by using basic algebraic rules. When dealing with rational exponents like \( a^{\frac{m}{n}} \), simplification usually focuses on reducing the fraction \( \frac{m}{n} \) to its simplest form. If \( m \) and \( n \) share a common factor, as in the example \( (3x^{2})^{\frac{4}{4}} \), the fraction reduces to 1, leaving us with \( (3x^{2})^{1} \).

Key Simplification Rules

• Any base raised to the power of one is the base itself: \( a^{1} = a \).
• Base powers can be multiplied by adding their exponents: \( a^{m} \cdot a^{n} = a^{m + n} \).
• Base powers can be divided by subtracting their exponents: \( \frac{a^{m}}{a^{n}} = a^{m - n} \).
• When raising a power to another power, multiply the exponents: \( (a^{m})^{n} = a^{m \cdot n} \).

In the exercise, simplifying \( \frac{4}{4} \) to 1 is an application of exponent simplification, leading to a more straightforward expression.

Properties of Exponents

The properties of exponents are rules that define how to manipulate expressions involving powers. These properties make it possible to simplify complex expressions and solve equations that would otherwise be challenging. A few fundamental properties of exponents were applied in our example to transform \( \sqrt[4]{(3x^{2})^{4}} \) into \( 3x^{2} \).

Understanding these properties is essential:

• The Product Rule: When multiplying like bases, add their exponents.
• The Quotient Rule: When dividing like bases, subtract the exponents of the base in the denominator from the exponent of the base in the numerator.
• The Power Rule: When raising a power to another power, multiply the exponents.
• The Zero Exponent Rule: Any base raised to the power of zero is equal to one (except when the base is also zero).
• The Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent, for instance, \( a^{-n} = \frac{1}{a^{n}} \).

By leveraging the properties of exponents, simplifying expressions becomes an achievable task, as shown by simplifying \( \frac{4}{4} \) down to 1 in the step-by-step solution.