Problem 103 - Elementary and Intermediate Algebra

Question

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[5]{\sqrt[3]{7 m}} $$

Step-by-Step Solution

Answer

The expression simplifies to \((7m)^{1/15}\).

Step 1: Express the innermost radical using rational exponents

The innermost radical is \( \sqrt[3]{7m} \). We express this radical using rational exponents: \( (7m)^{1/3} \).

Step 2: Express the outer radical using rational exponents

Now consider the entire expression \( \sqrt[5]{(7m)^{1/3}} \). We express this using rational exponents as: \( ((7m)^{1/3})^{1/5} \).

Step 3: Simplify using the power of a power property

To simplify \( ((7m)^{1/3})^{1/5} \), use the property of exponents: \((a^{m})^{n} = a^{m \cdot n}\). Here, \((7m)^{1/3 \cdot 1/5} = (7m)^{1/15}\).

Step 4: Final Simplification

The simplified form using rational exponents is \((7m)^{1/15}\). This expression no longer contains any radicals.

Key Concepts

Radical Simplification

Radical simplification is a process used to simplify expressions that involve roots. When you have multiple nested radicals, like \( \sqrt[5]{\sqrt[3]{7m}} \), you want to eliminate these radicals by expressing them as exponents. This helps streamline the expression and make it easier to work with.

To start with radical simplification, understand that the root symbol indicates an exponent. For instance, the cube root \( \sqrt[3]{x} \) can be rewritten using rational (or fractional) exponents as \( x^{1/3} \). Similarly, the fifth root \( \sqrt[5]{x} \) is \( x^{1/5} \). By expressing both the innermost and outermost radicals in these forms, you change the problem from dealing with roots to dealing with powers, which can be more straightforward.

Power of a Power Property

This property is fundamental when dealing with expressions involving exponents. The power of a power property states that when you raise a power to another power, you multiply the exponents. It is expressed as \( (a^m)^n = a^{m \cdot n} \).

In the context of the problem, once you've re-expressed both radicals in terms of exponents, you encounter an expression like \( ((7m)^{1/3})^{1/5} \). Here, both the \((7m)^{1/3}\) part and the \(^{1/5}\) exponent apply. Using the power of a power property, multiply these exponents:

• \( \frac{1}{3} \times \frac{1}{5} = \frac{1}{15} \)

Therefore, the entire expression simplifies to \((7m)^{1/15}\). This rule makes it easy to combine powers and simplify complex expressions.

Exponent Rules

Exponent rules are a set of guidelines that define how to work with powers effectively. These rules are vital for manipulating and simplifying expressions in algebra. Here are a few key rules you'll often use:

• Product of powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a product: \( (ab)^n = a^n b^n \)
• Power of a quotient: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \)

In our exercise, we specifically made use of the power of a power rule. This rule played a crucial role in simplifying the nested radicals into a single expression with a single rational exponent. Understanding and applying these rules allows you to tackle myriad problems involving exponents with confidence.