Problem 92 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \sqrt{\sqrt[3]{(3 x)^{2}}} $$

Step-by-Step Solution

Answer

\( (3x)^{1/3} \) is the simplified expression with rational exponents.

Step 1: Express the inner square root with rational exponents

The expression \( \sqrt[3]{(3x)^2} \) can be rewritten using rational exponents as \( (3x)^{2/3} \). This transformation follows from the rule that \( \sqrt[n]{a} = a^{1/n} \).

Step 2: Evaluate the outer square root

Apply the same logic of rational exponents to the outer square root. The expression \( \sqrt{(3x)^{2/3}} \) becomes \( ((3x)^{2/3})^{1/2} \). This is equivalent to raising \((3x)^{2/3}\) to the power of \(1/2\).

Step 3: Apply the power of a power rule

Apply the power of a power rule which states that \((a^m)^n = a^{m \cdot n}\). Here, \((3x)^{2/3}\) raised to the power of \(1/2\) becomes \((3x)^{(2/3) \cdot (1/2)} = (3x)^{1/3}\).

Step 4: Simplified expression

The expression is now simplified to \( (3x)^{1/3} \), which is expressed with rational exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra and calculus. It involves rewriting expressions in a more straightforward or more manageable form without changing their values. Understanding how to simplify can help make complex expressions easier to work with and can reveal underlying structures or properties that are not apparent at first glance.

In the context of the given exercise, we started with a nested radical expression \( \sqrt{\sqrt[3]{(3x)^2}} \). Simplification began by rewriting the inner radical \( \sqrt[3]{(3x)^2} \) using rational exponents, which made it more manageable. The entire process finishes by ensuring the final expression is in its simplest possible form, which was \( (3x)^{1/3} \).

The ability to simplify expressions effectively enhances mathematical understanding and problem-solving skills. It's always beneficial to be comfortable with these techniques, as it allows us to transform and manipulate expressions with ease.

Power of a Power Rule

The 'Power of a Power Rule' is a crucial principle in exponentiation. It states that when you raise an exponent to another exponent, you multiply the exponents. This can be expressed as:

• \((a^m)^n = a^{m \cdot n}\)

Applying this rule streamlines expressions dramatically, especially when working with nested exponential expressions.

In our example, after expressing both radicals with rational exponents, we ended up with \(((3x)^{2/3})^{1/2}\). To simplify further, we apply the 'Power of a Power Rule'. Thus, we multiply the exponents \(\frac{2}{3}\) and \(\frac{1}{2}\) to get \( (3x)^{1/3} \). This neat rule helps squeeze nested exponentials into a much simpler form, making calculations quicker and more efficient.

Often students may confuse this operation with adding exponents. Remember, multiplication of exponents only occurs when a power is raised to another power. Knowing when to use this rule is key in correctly transforming expressions.

Rational Exponent Rule

Rational exponents articulate roots as fractional powers. This rule turns complicated roots into much more tractable forms. The core rule to remember is:

• \(\sqrt[n]{a} = a^{1/n}\)

This is immensely helpful because it allows you to apply the laws of exponents to roots, making expressions easier to handle.

For instance, in the given problem, expressing \(\sqrt[3]{(3x)^2} \) as \((3x)^{2/3} \) is the application of this rule. The square root of the entire expression can also be represented as raising it to the power of \(1/2\). Consequently, using rational exponents simplifies the handling of nested roots significantly.

Understanding rational exponents also builds a strong foundation for more advanced topics in algebra and calculus. By practicing the transformation of radical expressions into exponents, students pave the way for solving more complex equations with ease. Emphasizing the consistency and continuity between radicals and rational exponents can demystify a lot of algebraic processes.