Problem 104 - Elementary and Intermediate Algebra

Question

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[3]{\sqrt[4]{21 x}} $$

Step-by-Step Solution

Answer

\((21x)^{\frac{1}{12}}\) is the simplified form.

Step 1: Understand the Expression

We need to simplify the expression \( \sqrt[3]{\sqrt[4]{21x}} \). This expression involves nested radicals, which means we will first convert each radical into an expression with a rational exponent.

Step 2: Convert the Inner Radical to Exponent Form

The inner radical is \( \sqrt[4]{21x} \). This radical can be rewritten with a rational exponent as \( (21x)^{\frac{1}{4}} \).

Step 3: Convert the Outer Radical to Exponent Form

The outer radical is \( \sqrt[3]{(21x)^{\frac{1}{4}}} \). To rewrite in exponent form, use \( (a^m)^n = a^{m \cdot n} \). So, \( (21x)^{\frac{1}{4}} \) raised to the power of \( \frac{1}{3} \) becomes \( (21x)^{\frac{1}{4} \cdot \frac{1}{3}} \).

Step 4: Simplify the Exponents

Next, simplify the expression \( (21x)^{\frac{1}{4} \cdot \frac{1}{3}} \) to \( (21x)^{\frac{1}{12}} \). This is done by multiplying the exponents: \( \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} \).

Step 5: Express the Simplified Form

The simplified expression is \( (21x)^{\frac{1}{12}} \). There are no further simplifications to perform, as this is already a simplified form of the original nested radicals.

Key Concepts

Simplifying Radicals

When we talk about simplifying radicals, we mean expressing them in a simpler or alternate form, usually by reducing or combining terms. Radicals can often look intimidating, but with the right approach, they become manageable.
In this exercise, involving the radical expression \( \sqrt[3]{\sqrt[4]{21x}} \), simplifying it begins with understanding its components.
**Steps to Simplify Radicals:**

• Identify the index of each radical. Here, \( \sqrt[3]{} \) is a cube root (with an index of 3) and \( \sqrt[4]{} \) is a fourth root (with an index of 4).
• Transform each radical into its equivalent expression using rational exponents. This makes calculations more straightforward.
• Carry out the necessary operations, such as multiplication or division, with these exponents to further simplify the expression.

By converting the radical into a rational exponent form and performing operations using exponent rules, simplifying becomes systematic and clear.

Nested Radicals

Nested radicals refer to a situation where one radical is inside another, like what we see in \( \sqrt[3]{\sqrt[4]{21x}} \). They can be complex but an effective strategy is breaking down these layers.
**Understanding Nested Radicals:**

• First, address the innermost radical. In our example, that's \( \sqrt[4]{21x} \), which converts to \( (21x)^{\frac{1}{4}} \).
• Then tackle the next outer radical layer. For the outer expression \( \sqrt[3]{(21x)^{\frac{1}{4}}} \), it can be rewritten as \( ( (21x)^{\frac{1}{4}} )^{\frac{1}{3}} \).
• Remember, \( (a^m)^n = a^{m \cdot n} \) which allows you to combine the powers into one single expression.

By methodically unraveling the nested radicals step-by-step and using exponent rules, this seemingly complex structure can be simplified into a much more manageable form.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. In our problem, \( 21x \) is the algebraic expression under the radicals.
To simplify or manipulate algebraic expressions within radicals, it's crucial to:

• Recognize the components, such as constants (like 21) and variables (like \( x \)).
• Apply algebraic rules and exponent laws that can simplify the expression when under a radical or converted to rational exponents.
• Maintain the solution accurately by handling variables with care, especially when they are part of more complex expressions like nested radicals.

Combining these principles with systematic approaches ensures that even when dealing with radicals, the algebraic expressions remain straightforward and less daunting.